Enumeration of general planar hypermaps with an alternating boundary

Enumeration of general plana r hyp ermaps with an alternating b ounda ry V alen tin Baillard 1,2 , Ariane Carrance 3 , and Bertrand Eynard 1,4 1 Univ ersité P aris-Saclay , CNRS, CEA, Institut de physique théorique, 91191, Gif-sur-Y vette, F rance. 2 Institut de Mathématiques d’Orsay , Université P aris-Sacla y , 91400, Orsay , F rance. 3 F akultät für Mathematik, Universität Wien, 1090 Wien, Österreich. 4 CRM, Centre de recherc hes mathématiques de Montréal, Montréal, QC, Canada. Marc h 31, 2026 In this pap er, w e extend the en umerative study of planar hypermaps with an alternating b oundary in tro duced in [BC21] . In that article, an explicit rational parametrization was obtained for the asso ciated generating function in the case of m -constellations, using a v ariant of the k ernel method. W e dev elop here a new strategy to obtain an algebraic equation in the general case, whic h includes maps decorated by the Ising mo del, through a classical man y-to-one corresp ondence. One of the main steps of our strategy is the sim ultaneous elimination of tw o catalytic v ariables. W e then apply this strategy to the case of Ising quadrangulations, where w e obtain an explicit rational parametrization. As a consequence, w e sho w that some notable properties of the constellations case are no longer satisfied in general. Contents 1. Intro duction 2 1.1. Con text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2. Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2. Reminders on related w o rks 6 2.1. Hyp ermaps with a mono c hromatic b oundary . . . . . . . . . . . . . . . . . . . . . . . 6 2.2. Previous results on the alternating b oundary condition . . . . . . . . . . . . . . . . . . 7 3. Preliminaries 9 3.1. General definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.2. Corresp ondence with the Ising mo del . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.3. T utte decomp ositions and splitting pro cedure . . . . . . . . . . . . . . . . . . . . . . . 11 4. Master equation for the alternating b ounda ry condition 12 4.1. Splitting equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1 4.2. Algebraicit y of b f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.3. Re-deriv ation of the T utte equation and k ernel system . . . . . . . . . . . . . . . . . . 18 5. Explicit solution for Ising quadrangulations 19 5.1. P arametrization for the mono chromatic b oundary . . . . . . . . . . . . . . . . . . . . . 19 5.2. Explicit master equation for b f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 5.3. The symmetric case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 6. Conclusion and p erspectives 26 A. Application of the k ernel metho d to general potentials 26 1. Intro duction 1.1. Context The en umerativ e theory of planar maps has b een an active topic in combinatorics since its inception by T utte in the sixties, through a series of seminal pap ers, including [T ut62a; T ut62b; T ut63; T ut68] . An accoun t of some of its more recen t dev elopments ma y b e found in the review b y Schaeffer [Sc h15] . Many suc h developmen ts were motiv ated b y connections with other fields: theoretical physics, algebraic and en umerative geometry , computer science, probability theory ... In the physics literature, the problem of enumerating planar maps emerged in connection with matrix mo dels: this w as initiated b y t’Ho oft [Ho o74] and further dev elop ed in particular by Brezin, Itzykson, P arisi and Zub er [BIPZ78] . A notable motiv ation comes from t wo-dimensional quantum gra vity , see [FGZJ95] for a detailed review. In the language of ph ysics, usual maps provide either a mo del of pure gra vity without matter, or with some v ery restricted type of matter. It is therefore natural to consider more generally maps endow ed with a mo del of statistical mechanics. In particular, w e consider here hypermaps, which are maps whose faces are bicolored (sa y , in black and white) in such a wa y that no tw o faces of the same color are adjacent. They are related through a classical many- to-one corresp ondence to maps endo wed with an Ising mo del on their faces (see Section 3.2 ). The en umeration of hypermaps was th us first addressed in the physics literature through the tw o-matrix mo del [IZ80; Meh81; Dou91; Sta93; DKK93] , leading to the iden tification of the critical exp onents of the Ising mo del on random maps in [Kaz86; BK87] . While it is fully rigorous to equate the correlation functions of formal matrix mo dels with generating functions of maps (see for instance [Eyn11] ), the connection with c onver gent matrix mo dels is muc h more subtle (see [GMS06] ). T o en umerate maps, the standard approach consists in studying the effect of the remov al of an edge. Now adays, this op eration is often called “p eeling”, see for instance [Cur23] . In this pap er, w e also use the term splitting, that comes from matrix mo dels, see Section 3.3 . In order to turn the p eeling in to equations, one needs to keep trac k of one or more auxiliary parameters, called c atalytic variables [BMJ06] , and typically corresp onding to b oundary lengths. In the context of hypermaps, a further complication o ccurs, since one needs to keep trac k of b oundary conditions: these enco de the colors of the faces inciden t to the b oundary . The most tractable b oundary condition is the so-called Dobrushin b oundary condition, which consists in having the b oundary made of t wo parts: one part is inciden t to white faces, and the other to black faces. The Dobrushin b oundary condition has the k ey prop ert y that it is inv arian t under p eeling, pro vided that we alw ays p eel an edge at the white- blac k interface. The resulting equations are solved in [Eyn03; Eyn16] and in [CT20] . Kno wing how to treat Dobrushin b oundary conditions, one may then consider “mixed” b oundary conditions [EO05] where there is a prescrib ed n umber of white-blac k interfaces. How ever, this approach seems to b ecome 2 in tractable when the num b er of in terfaces gets large. An earlier w ork b y Bouttier and the second author [BC21] in tro duced the alternating b oundary condition, whic h corresp onds to the extreme situ- ation where the num b er of interfaces is maximal, that is to say when white and blac k faces alternate along the b oundary . Note that [EO05; EO08; Eyn16] also treats the case of mixed b oundary conditions in higher genera and with more than one b oundary , through the formalism of top olo gic al r e cursion . While the present pap er only deals with the disk top ology , w e will touc h up on the notion of sp e ctr al curve , as constructed in top ological recursion, when discussing prop erties our generating functions, see Section 2.2 . It is natural to conjecture that, like many other map en umeration problems, h yp ermaps with an alternating b oundary also fall under the general framework of top ological recursion. W e discuss this in more detail in Sections 2.2 and 6 . Bey ond the in teresting com binatorial phenomena that we uncov er in the present pap er, there are some external motiv ations to study the alternating b oundary condition. First, the initial motiv ation of [BC21] came from the probabilistic study of bicolored triangulations. Indeed, enumeration asymp- totics of the alternating b oundary condition obtained in [BC21] were a crucial ingredient in a related w ork by the second author [Car21] , that prov ed that large random uniform bicolored triangulations con verge to the Br ownian spher e . In the case of the Ising mo del, the alternating b oundary condition is also natural to consider when studying the antiferr omagnetic regime. Indeed, for bipartite lattices, one can transform an antiferromagnetic mo del into a ferromagnetic one, by flipping the spins of o dd sites (see for instance [FV18, Section 3.10.5] ), and this transforms the alternating boundary condi- tion into the classical mono chromatic one. Moreov er, the Ising mo del on random maps is related to Calo ger o-Moser spin chains by a bulk/b oundary corresp ondence, see for instance [EKR15, Section 6.1] . Through this correspondence, the partition function of the alternating condition is related to disor der op er ators of the spin c hain. Thus, com binatorial results on the alternating condition open up new directions in the study of these statistical mec hanics mo dels. Before presenting our main results, let us mention that there exist many bijectiv e approaches to the enumeration of hypermaps. While most of them fo cus on the mono chromatic b oundary [BMS03; BDF G04; BDF G07; BFG14; BF12; AMT25] , the Dobrushin b oundary condition was recently recov ered using the bijective metho d of the slice decomp osition [AB25] . W e will address the adaptation of the slice decomp osition to the alternating b oundary in a future work [CEL] . 1.2. Main results Recall that a planar map is a connected multigraph dra wn on the sphere without edge crossings, and considered up to con tin uous deformation. It consists of v ertices, edges, faces, and corners. A b oundary is a distinguished face, whic h we often choose as the infinite face when drawing the map on the plane. It is assumed to b e r o ote d , that is to sa y there is a distinguished corner along the b oundary , depicted as gra y arro w in our illustrations. The other faces are called inner fac es . A hyp ermap with a b oundary is a planar map with a b oundary , where ev ery inner face is colored either black or white, in a such a w ay that adjacent inner faces hav e different colors. Note that the b oundary do es not ha ve a sp ecified color: it may b e incident to faces of b oth colors. Let w b e a word on the alphab et {• , ◦} , and ℓ be the length of w. A h yp ermap with b oundary condition w is a hypermap with a b oundary of length ℓ such that, when walking in clo ckwise direction around the map (starting at the ro ot corner), 1. if the i -th visited edge ( i = 1 , . . . , ℓ ) is incident to an inner face, then this face is white if w i = • and black if w i = ◦ , 2. if an edge is inciden t to the b oundary on b oth sides (i.e. it is a b oundary bridge ), and is then visited twice, sa y at steps i and j, then w i  = w j . 3 In other words, w i co des the color of the b oundary face as seen from the i -th edge. Remark 1.1. L o oking at a given hyp ermap with a b oundary, ther e is a p ossible ambiguity for the asso ciate d b oundary c ondition if ther e ar e bridges (se e Figur e 1 ), but this c auses no issue for our gener ating functions, as they c ount hyp ermaps that admit given b oundary c onditions. Figure 1: Left, a hyp ermap that admits the b ounda ry conditions ◦ • ◦ • • • ◦ • • • ◦ ◦ ◦ and ◦ • ◦ • • ◦ ◦ • • • ◦ • ◦ . Right, a hyp ermap with a mono chromatic b oundary . A hyp ermap with a white (r esp. black) mono chr omatic b oundary is a h yp ermap with b oundary condition ◦ ◦ · · · ◦ (resp. • • · · · • ). These are just hypermaps in the sense of [BF12] . A hyp ermap with an alternating b oundary is a hypermap with b oundary condition ◦ • ◦ • · · · ◦ • . Note that a hypermap with a mono chromatic b oundary cannot ha ve a b oundary bridge, but ma y hav e pinc h points (see Figure 1 ). F or a hypermap with an alternating b oundary , bridges are p ossible, but they must necessarily hav e blac k neigh b oring b oundary edges on one side, and white ones on the other (see Figure 2 ). Throughout this pap er, we consider generating functions for h yp ermaps whose white (resp. black) face degrees are upp er b ounded b y some integer d (resp. e d ). W e will keep trac k of the num b er of faces of eac h color and degree with v ariables t 2 , . . . , t d , e t 2 , . . . , e t e d . The follo wing p olynomials in Q [ t 2 , . . . , t d ][ x ] (resp. Q [ e t 2 , . . . , e t e d ][ y ] ): V ( x ) = − X 2 ≤ i ≤ d t i i x i , e V ( y ) = − X 2 ≤ i ≤ e d e t i i y i (1) will b e useful quantities, that w e call p otentials b y analogy with matrix mo dels. Figure 2: Left, a hyp ermap with an alternating boundary . Right, a hyp ermap whose b oundary is not alternating. Let F r denote the generating function of hypermaps with an alternating b oundary of length 2 r , coun ted with a w eight t p er vertex, weigh t c − 1 p er edge, t i p er white inner face of degree i , and e t j p er 4 blac k inner face of degree j . By conv ention, F 0 = 1 (corresp onding to the trivial vertex map). Our main series of interest is b f ( ω ) = b f ( t, c, t 2 , . . . , t d , e t 2 , . . . , e t e d ; ω ) := X r ≥ 0 F r ω r +1 − c. (2) The extra − c migh t seem arbitrary , but it will help simplify some formulae in what follo ws. W e define similarly W p as the generating function of hypermaps with a white mono c hromatic b oundary of length p , with the same weigh ts, and W ( x ) = W ( t, c, t 2 , . . . , t d , e t 2 , . . . , e t e d ; x ) := X r ≥ 0 W p ω p +1 , Y ( x ) := 1 c ( W ( x ) − V ′ ( x )) (3) W e are now ready to state our main results. Theorem 1.1. F or any d, e d ≥ 2 , b f is algebr aic over Q (( t ))( c, t 1 , . . . , t d , e t 1 , . . . , e t e d , ω ) . Mor e over, we have an explicit str ate gy to obtain its annihilating p olynomial, that do es not r ely on the kernel metho d or its gener alizations, se e The or em 4.4 for a mor e pr e cise statement. Note that it was mentioned in [BC21] that the general case is also amenable to the kernel metho d. W e give details on this in App endix A , and compare it with our strategy in Section 4.2 . In particular, in the case of Ising quadrangulations, our strategy is muc h simpler than the kernel method, and leads to the following: Theorem 1.2. F or the sp e cialization t 2 = e t 2 , t 4 = e t 4 , t k = e t k = 0 if k  = 2 , 4 , i.e. symmetric Ising quadr angulations, we have the fol lowing explicit r ational p ar ametrization of ( ω , b f ( ω )) in terms of a formal variable h : ω sym ( h ) = ( α 3 h + γ ) 2 γ 3 h ( γ h + α 3 ) 2 α 3 b f sym ( h ) = − c ( α 3 γ h 2 + h ( α 2 1 − 2 α 3 γ ) + α 3 γ )( γ h + α 3 ) 3 ( α 3 h + γ ) γ 4 ( h − 1) 2 h 2 , (4) wher e γ , α 1 , α 3 ar e algebr aic functions in c, t 2 , t 4 , t that ar e determine d by t 4 = cα 3 γ 3 , t 2 = cα 1 γ − 3 α 3 γ 2 , t = c ( γ 2 − α 2 1 − 3 α 2 3 ) . One remark able feature of the case of m -constellations that was solv ed in [BC21] , is that ( ω , b f ) can b e parametrized as rational functions of ( x, Y ( x )) , that satisfy the kernel r elation ω b f ( ω ) + cxY ( x ) = 0 (see the detailed discussion in Section 2.2 ). As a consequence of Theorem 1.2 , we show that this no longer holds in the general case: Corollary 1.3. In gener al, the curves ( ω , b f ( ω )) and ( x, Y ( x )) do not admit a joint r ational p ar ametriza- tion that satisfies the kernel r elation. The rest of the pap er is organized as follows. In Section 2 , we recall some imp ortant results from previous w orks. In Section 2.1 , w e recall properties of the monochromatic boundary from [Eyn16] and in tro duce the biv ariate p olynomial E ( x, y ) called the sp ectral curve, and in Section 2.2 , w e recall some relev ant results from [BC21] and discuss some consequences that were not stated there. In Section 3 , w e set up the notation for all the auxiliary functions that w e will need, and recall the corresp ondence 5 b et ween hypermaps and the Ising mo del on maps. W e also establish a T utte decomp osition/splitting pro cedure for general b oundary conditions in Prop osition 3.2 . In Section 4 , we apply Prop osition 3.2 to v arious auxiliary functions, leading to an equalit y b et ween the sp ectral curv e E ( x, y ) that do es not dep end on ω , and another biv ariate p olynomial Q ( x, y ) whose co efficients are explicit functions of ω , b f ( ω ) and some other auxiliary generating functions, as stated in Prop osition 4.3 . Since b oth p olynomials hav e the same degree, same leading coefficient and v anish on ( x, Y ( x )) , they m ust be equal, and requiring the v anishing of coefficients of Q − E thus giv es a p olynomial system on b f ( ω ) and related functions. W e then prov e that this system has non-zero Jacobian determinant ev aluated at those functions, which gives Theorem 4.4 . W e also compare this strategy to the kernel metho d. W e the n show in Proposition 4.5 that we reco ver the T utte equation of [BC21] from our splitting equations. In Section 5 , we apply the strategy of Theorem 4.4 : we obtain an explicit master equation in the general (non-symmetric) case in Section 5.2 , and solve it for the symmetric case in Section 5.3 , yielding Theorem 1.2 . W e conclude with a list of op en questions in Section 6 . In App endix A , we give details on the application of the k ernel metho d to the T utte equation of [BC21] for general p otentials. 2. Reminders on related w o rks 2.1. Hyp ermaps with a mono chromatic b oundary Let us recall some imp ortan t results from [Eyn03] summarized in [Eyn16, Chapter 8] . Note that w e are almost using the exact same con v ention, sa v e for the fact that we consider proper hypermaps, while [Eyn16, Chapter 8] describ es the Ising mo del via bicolored digons in maps with mono chromatic edges. The only difference that this entails is that it introduces in our formualae a minus sign in fron t of the parameter c that counts edges. Hence we define Y ( x ) = 1 c ( W ( x ) − V ′ ( x )) , where W ( x ) = W (0) 1 ( x ) in [Eyn16] . It is an algebraic function of x, t, { t i } , { e t j } , and ( x, Y ( x )) admits the follo wing explicit rational parametrization: Theorem 2.1 (Theorem [Eyn03] and 8.3.1 in [Eyn16] ) . F or a formal variable z , we have x ( z ) = γ z + e d X k =0 α k z − k , Y ( x ( z )) = y ( z ) = γ z − 1 + d X k =0 β k z k , (5) wher e the c o efficients α k , β k , γ ar e algebr aic functions of t, { t i } , { e t j } , that ar e char acterize d by: V ′ ( x ) + cy = t γ z + O ( z − 2 ) , ˜ V ′ ( y ) + cx = tz γ + O ( z 2 ) , γ 2 = − ct t 2 e t 2 − c 2 + O ( t 2 ) . (6) Equiv alently , Y ( x ) is characterized as the unique p ow er series solution of E ( x, y ) = 0 , where the 6 p olynomial E ( x, y ) is equal to (see [Eyn16, Theorem 8.3.2] ): E ( x, y ) = − ( − 1) d c 2 γ d + ˜ d − 2 det                  γ α 0 − x α 1 . . . α e d − 1 γ α 0 − x α 1 . . . α e d − 1 . . . . . . . . . . . . γ α 0 − x α 1 . . . α e d − 1 β d − 1 . . . β 1 β 0 − y γ β d − 1 . . . β 1 β 0 − y γ . . . . . . β d − 1 . . . β 1 β 0 − y γ                  . (7) It can also b e written as E ( x, y ) = ( V ′ ( x ) + cy )( ˜ V ′ ( y ) + cx ) + 1 c P ( x, y ) − ct, (8) where P ( x, y ) =  V ′ ( x ) − V ′ ( A ) x − A ˜ V ′ ( y ) − ˜ V ′ ( B ) y − B  . (9) 2.2. Previous results on the alternating b ounda ry condition Let us recall and revisit some results from [BC21] . This work established a T utte equation (equation (27) there) on the generating function that w as called M ( x, w ) there, and that we denote here by M ( x, w ) to av oid ambiguit y . As the notational conv entions of [BC21] are sligh tly differen t from those w e use here (see Section 4 for more details), this generating function is related to the ones of the presen t pap er by M ( x, w ) = 1 w M  x, 1 w  − W ( x ) , and the generating function of interest in [BC21] is A ( w ) = tw + M (0 , w ) = f 00 (1 /w ) . Then, equation (27) in [BC21] is written as K ( w , x ) M ( w, x ) = R ( w , x ) , (10) with K ( x, w ) = 1 − A ( w ) − w xY ( x ) , R ( x, w ) = w xW ( x ) Y ( x ) − w X i t i  x i ( M ( x, w ) + W ( x ))  x ≥ 0 . W e compare this equation with the functional equations that w e derive in the present pap er in Sec- tion 4.3 . Let us fo cus for the moment on ( 10 ) and its role in [BC21] . In full generality , the remainder term R is hard to tac kle. How ever, it was mentioned in [BC21, Section 3] that ( 10 ) can b e rewritten in a form where one can apply [BMJ06, Theorem 3] , and conclude that M ( x, w ) , hence A ( w ) , is algebraic in w, x, W ( x ) , t, t i , e t j . As this argument was not detailed in [BC21] , and w e develop here another path to the algebraicit y of A ( w ) , we detail in App endix A ho w ( 10 ) can be rewritten in suc h a form, so that w e can compare the tw o strategies in detail in Section 4.2 . 7 In the sp ecial case of m -c onstel lations (i.e. for the sp ecialization t k = 0 for k  = m and e t ℓ = 0 if m  | ℓ ), R ( x, w ) simplifies greatly (see [BC21, Section 4] ): R ( x, w ) = w xY ( x ) 2 − wx m Y ( x ) − x m − 1 A ( w ) . Going back to equation ( 10 ), one can then p erform a v ariant of the kernel metho d (see for in- stance [BMJ06] for a detailed exp osition, and App endix A for a more classical application of it to solv e ( 10 ) in the general case). Indeed, instead of eliminating the c atalytic variable x from ( 10 ), one can consider the unique formal p ow er series w ( ξ ) in ξ − 1 suc h that w ( ξ ) = 1 − A ( w ( ξ )) ξ . (11) Since the substitution M ( x, w ( xY ( x ))) is w ell-defined, b y ( 10 ) we deduce that R ( x, w ( xY ( x ))) = 0 as well. This gives a linear system of t wo equations on the series w ( xY ( x )) and A ( w ( xY ( x ))) , whic h determines them as Theorem 2.2 (Explicit parametrization for m -constellations, Theorem 1 in [BC21] ) . w ( xY ( x )) = x m ( xY ( x )) 2 , A ( w ( xY ( x ))) = 1 − x m xY ( x ) . One might worry that these form ulae do not dep end only on xY ( x ) , but also on x m . Ho wev er, xY ( x ) is actually a Laurent series in x − m : xY ( x ) = Ξ( x m ) = x m + t + ... , and one can p erform a Lagrange in version, to express an y analytic function in x − m , as an analytic function in ( xY ( x )) − 1 . Thus, the ab o ve form ulae prop erly define the functions w (Ξ) , A (Ξ) . One can p erform the additional substitution x = x ( z ) , or more precisely Ξ( x m ) = ξ ( z m ) , with ξ ( z m ) = x ( z ) y ( z ) b eing a uniquely defined Laurent p olynomial in z m . Since χ ( z m ) = x ( z ) m is also w ell-defined as a Laurent polynomial in z m , this yields the rational parametrization w ( s ) = χ ( s ) ξ ( s ) 2 , a ( s ) = 1 − χ ( s ) ξ ( s ) , (12) whic h w as stated in Theorem 1 in [BC21] . Th us, in the case of m -constellations, the algebraic curve ( w , A ( w )) can b e expressed as a pair of rational functions ov er the algebraic curve ( x ( z ) , y ( z )) corresp onding to the mono chromactic b ound- ary condition. Moreo ver, this parametrization of ( w , A ( w )) ov er ( x ( z ) , y ( z )) satisfies the kernel rela- tion ( 11 ). Let us no w discuss another remark able feature of this parametrization, that was not stated in [BC21] . Let us switch the v ariable w to the inv erse v ariable ω (as is the case in the present pap er), and write Ω( s ) for w ( s ) − 1 . Geometrically , the mapping φ : CP 1 → CP 1 , φ ( z ) = z m is a branc hed co vering of degree m . T o the curve ( x ( z ) , y ( z )) (and its explicit parametrization), one asso ciates a canonical symplectic form σ = d x ∧ d y – note that this has deep er implications as this curve is upgraded to a sp e ctr al curve through the formalism of top ological recursion, when one generalizes the en umeration problem to more general top ologies, see [Eyn16] . Likewise, w e define 𝜏 = dΩ ∧ d a from (Ω( s ) , a ( s )) – here the enumeration problem for other top ologies has yet to b e solved, and thus one cannot really talk of a sp ectral curv e for now, see the discussion in Section 6 . Then, we hav e the following prop erty: 8 Prop osition 2.3. W e have φ ∗ 𝜏 = m · d x ∧ d y , i.e. the pul lb ack of 𝜏 by φ is exactly e qual to σ multiplie d by the de gr e e of the c overing φ . Pr o of. W e hav e φ ∗ 𝜏 = d(Ω ◦ φ ) ∧ d( a ◦ φ ) = d  y 2 x m − 2  ∧ d  1 − x m − 1 y  = 1 x m − 1  2 xy d y − ( m − 2) y 2 d x  ∧ 1 y 2  − ( m − 1) y x m − 2 d x − x m − 1 d y  = m · d x ∧ d y . Giv en this v ery nice geometric prop erty , it is natural to wonder whether, when considering higher genera, the generating functions of the alternating b oundary condition can also b e expressed nicely in terms of the ones of the mono c hromatic condition. If this is the case, this would b e analogous to the symple ctic invarianc e phenomenon, that occurs for sp ectral curv es related b y symplectomorphisms (see for instance [BCGF24] ). These properties of the m -constellation case w ere a strong motiv ation to study the alternating b oundary condition on more general h yp ermaps. In this pap er, we give some partial answers for the general case for the disk top ology . In particular, w e show that on more general hypermaps, ( ω , A ( ω − 1 )) cannot b e written as rational functions of ( x, y ) satisfying the kernel relation ( 11 ), see Corollary 1.3 . 3. Prelimina ries 3.1. General definitions As we will handle auxiliary generating functions with v arious types of b oundary conditions, it will b e con venien t to b orrow some notational conv entions from formal matrix mo dels. Definition 3.1. L et d ∈ N , e d ∈ N and let P b e a wor d over { A, B } : we identify such a wor d with a non- c ommutative monomial in A, B , and as a b oundary c ondition for hyp ermaps thr ough the c orr esp ondenc e A 7→ • , B 7→ ◦ . W e denote by Bic d, e d ( P ) the set of hyp ermaps of the disk with b oundary c ondition P , and whose white (r esp. black) fac es ar e of de gr e e at most d (r esp. e d ). W e then define  P  := X m ∈ Bic d, e d ( P ) t v ( m ) c − e ( m ) Y 2 ≤ i ≤ d t f i ◦ ( m ) i Y 2 ≤ j ≤ e d e t f j • ( m ) j , wher e v ( m ) is the numb er of vertic es of m , e ( m ) its numb er of e dges, and f i □ ( m ) its numb er of fac es of c olor □ and de gr e e i . In other wor ds,  P  is the gener ating function of maps in Bic d, e d ( P ) , enumer ate d with r esp e ct to their numb er of vertic es, e dges, and fac es of given de gr e e. W e extend this definition by line arity first to any non-c ommutative p olynomial in A, B , then to any formal series in the non-c ommutative variables A and B . W e take the c onvention that Bic d, e d (1) = {·} (the trivial vertex map) and thus  1  = t , and that  ∅  = 1 (and Bic d, e d ( ∅ ) = ∅ ). 9 Remark 3.1. Note that line arity al lows to write gener ating series such as 1 x − A = ∞ X k =0 x − k − 1 A k Thus, our main gener ating function b f is e qual to: b f = 1 ω X k ≥ 0 1 ω k  ( AB ) k  − c =  1 ω − B A  − c. W e define similarly the following auxiliary generating functions: f ij ( ω ) :=  A i 1 ω − B A B j  , D ij :=  A i B j  for i, j ∈ Z ≥ 0 W ( x ) :=  1 x − A  , M ( x, ω ) :=  1 x − A 1 ω − B A  , R ( x, ω ) :=  V ′ ( x ) − V ′ ( A ) x − A 1 ω − B A  , S + ( x, y , ω ) :=  V ′ ( x ) − V ′ ( A ) x − A 1 ω − B A 1 y − B  , b P ( x, y , ω ) :=  V ′ ( x ) − V ′ ( A ) x − A 1 ω − B A e V ′ ( y ) − e V ′ ( B ) y − B  , (13) and, for functions that are not symmetric in A ↔ B and x ↔ y , we denote the mirrored function with a tilde ( f W ( y ) , etc.). W e will also make use the generating functions of the mono chromatic b oundary condition that were defined in Section 2.1 : cY ( x ) = W ( x ) − V ′ ( x ) , cX ( y ) = f W ( y ) − e V ′ ( y ) . 3.2. Co rresp ondence with the Ising mo del Let us no w turn to the connection b etw een h yp ermaps and the Ising mo del on maps, i.e. maps carrying +/- spins on their faces. A word P on { A, B } can also b e seen as a w ord ov er { + , −} , and th us as a b oundary condition for Ising maps. Let I d, e d ( P ) b e set of Ising planar maps with b oundary condition P and + (resp. − ) faces of degree at most d (resp. e d ). Let us define the generating function of I d, e d ( P ) as I ( P ) = X m ∈ I d, e d ( P ) t v ( m ) Y 3 ≤ i ≤ d t f i + ( m ) i Y 3 ≤ j ≤ e d e t f j − ( m ) j c e ++ ( m ) ++ c e −− ( m ) −− c e + − ( m ) + − , where e □ △ ( m ) is the num b er of edges of type □ △ in m . Then we hav e Prop osition 3.1. With the matching of p ar ameters  c 2 − ct 2 − c e t 2 c 2  =  c + − c −− c ++ c + −  − 1 , we have, for any b oundary c ondition P ,  P  = I ( P ) . Pr o of. Starting from a h yp ermap m ∈ Bic d, e d ( P ) , let us con tract its digons (see Figure 3 ) and in terpret the colors of the remaining faces as spins. This is a man y-to-one corresp ondence from Bic d, e d ( P ) to I d, e d ( P ) . If t wo h yp ermaps m 1 , m 2 yield the same Ising map m , necessarily b oth m 1 and m 2 are obtained from m b y inserting on eac h edge a sequence of digons that alternate in color, and suc h that the initial 10 + − − + − − + − − + + Figure 3: Illustration of the many-to-one co rresp ondence, from hyp ermaps to Ising-decorated maps, through the contraction of digons. and final digons hav e the color opp osite of the face of m that they are glued to. Therefore, for each edge of m , b oth the n umber of white and black digons in b oth sequences m ust hav e the same parity . This implies that  P  = X m ∈ I d, e d ( P ) t v ( m ) Y 3 ≤ i ≤ d t f i + ( m ) i Y 3 ≤ j ≤ e d e t f j − ( m ) j c e t 2 c 2 − t 2 e t 2 ! e ++ ( m )  ct 2 c 2 − t 2 e t 2  e −− ( m )  c 2 c 2 − t 2 e t 2  e + − ( m ) , whic h matc hes I ( P ) provided that  c 2 − ct 2 − c e t 2 c 2  =  c + − c −− c ++ c + −  − 1 . 3.3. T utte decomp ositions and splitting procedure W e now prov e a T utte-type decomp osition for the generating function  P  for an y giv en b oundary condition P . W e call such a decomp osition, a splitting pr o c e dur e , once again by analogy with matrix mo dels, see for instance [Eyn11; Eyn16] . Prop osition 3.2. L et P b e a wor d over { A, B } , then c  P A  = −  P e V ′ ( B )  + X B in P s.t. P = P 1 B P 2  P 1  P 2  . Pr o of. Consider a hypermap m with b oundary condition P A , contributing to h t v c − e Y 2 ≤ i ≤ d t f i ◦ i Y 2 ≤ j ≤ e d e t f j • j i  P A  , 11 and p eel the edge asso ciated to A on the b oundary of m (see Figure 4 ). If it was incident to an inner blac k face of degree k , the map m ′ obtained b y remo ving that face has one less inner blac k face of degree k , one less edge, and k − 1 more b oundary edges of type B , that are inserted in the b oundary condition where A was. Thus, m ′ con tributes to h t v c − e +1 Y 2 ≤ i ≤ d t f i ◦ i · e t f k • − 1 k Y j  = k e t f j • j i  P B k  . If A was iden tified with a b oundary edge of t yp e B , then m is made of tw o smaller maps m 1 , m 2 connected by a bridge, and their resp ective boundary conditions P 1 , P 2 m ust satisfy P = P 1 B P 2 . Moreo ver, their num b er of edges sum up to e − 1 , while all the inner faces of m app ear in either m 1 or m 2 . Therefore, the pair ( m 1 , m 2 ) contributes to h t v c − e +1 Y 2 ≤ i ≤ d t f i ◦ i Y 2 ≤ j ≤ e d e t f j • j i  P 1  P 2  . Summing ov er all p ossible maps, and as X 2 ≤ k ≤ e d e t k B k = − e V ′ ( B ) , w e obtain the announced equation. F rom our definition of  ·  , this extends linearly to any formal non-comm utative p ow er series in A, B : if P = P k c k w k , then c  P A  = −  P e V ′ ( B )  + X k c k X B in w k s.t. w k = P 1 B P 2  P 1  P 2  . (14) W e also ha ve the symmetric equation for the splitting of  P B  . Remark 3.2. Note that P 1 or P 2 in the sum c an b e 1 . Also, note that the sum c an b e empty (if ther e is no p ower of B in P ). 4. Master equation fo r the alternating b ounda ry condition 4.1. Splitting equations W e will now derive tw o equations on the function M , inv olving the other auxiliary generating functions, b y applying the splitting pro cedure to v arious series. Prop osition 4.1. The gener ating function M satisfies the fol lowing two e quations: c ( y − Y )( x e S + − e R ) = M  cx ( e V ′ + cx − f 10 ) + e Rω  − c 2 xf 00 + x b P + e R ( − V ′ − cy + f 01 ) , (15) M ( b f ω + cxY ) = − xR + b f ( W − f 01 ) . (16) 12 P A P B k − 1 P 1 P 2 A B P 1 P 2 A B Figure 4: A sketch of the splitting procedure: top left, the b ounda ry edge of type A that w e p eel. T op right, the case where it is adjacent to a black inner face. Bottom row, the case where it is identified with a b oundary edge of type B . Pr o of. Let us first write  1 x − A 1 ω − B A B e V ′ ( y ) − e V ′ ( B ) y − B  =  1 x − A 1 ω − B A ( e V ′ ( y ) − e V ′ ( B ))  y y − B − 1   = y  1 x − A 1 ω − B A e V ′ ( y ) − e V ′ ( B ) y − B  −  1 x − A 1 ω − B A ( e V ′ ( y ) − e V ′ ( B ))  = y e S + − e V ′ M +  1 x − A 1 ω − B A e V ′ ( B )  = y e S + − e V ′ M −  c  1 x − A 1 ω − B A A  −  1 x − A 1 ω − B A  1 ω − B A A   " ↑ Splitting of  1 x − A 1 ω − B A A  # = y e S + − e V ′ M − c   x x − A − 1  1 ω − B A  + M f 10 = y e S + − e V ′ M − cxM + cf 00 + M f 10 . 13 On the other hand: c  1 x − A 1 ω − B A B e V ′ ( y ) − e V ′ ( B ) y − B  =  1 x − A 1 ω − B A V ′ ( A ) e V ′ ( y ) − e V ′ ( B ) y − B  +  1 x − A  1 x − A 1 ω − B A e V ′ ( y ) − e V ′ ( B ) y − B  +  1 x − A 1 ω − B A B  1 ω − B A e V ′ ( y ) − e V ′ ( B ) y − B  " ↑ Splitting of c  1 x − A 1 ω − B A B e V ′ ( y ) − e V ′ ( B ) y − B  # = − V ′ ( x ) e S + + b P + W e S + +  1 x − A 1 ω − B A B  e R = − V ′ ( x ) e S + + b P + W e S + +  1 x  1 + A x − A  1 ω − B A B  e R = − V ′ ( x ) e S + + b P + W e S + + e R x  f 01 +  1 x − A 1 ω − B A B A   = − V ′ ( x ) e S + + b P + W e S + + e R x  f 01 +  1 x − A  ω ω − B A − 1    = − V ′ ( x ) e S + + b P + W e S + + e R x ( f 01 + ω M − W ) . Com bining the tw o previous equations, we obtain: c  y e S + − e V ′ M − cxM + cf 00 + M f 10  = − V ′ ( x ) e S + + b P + W e S + + e R x ( f 01 + ω M − W ) , whic h can b e rewritten as e S + ( cy + V ′ − W ) = M c e V ′ + c 2 x − cf 10 + e Rω x ! − c 2 f 00 + b P + e R x ( f 01 − W ) , whic h yields equation ( 15 ). Let us now turn to ( 16 ). W e hav e: c  1 x − A 1 ω − B A B  = −  1 x − A 1 ω − B A V ′ ( A )  +  1 x − A  1 x − A 1 ω − B A  +  1 x − A 1 ω − B A B  1 ω − B A  " ↑ Splitting of c  1 x − A 1 ω − B A B  # = − V ′ M + R + W M +  1 x − A 1 ω − B A B  f 00 , so that  1 x − A 1 ω − B A B  = − 1 b f ( R + cY M ) . 14 On the other hand,  1 x − A 1 ω − B A B  =  1 x  1 + A x − A  1 ω − B A B  = 1 x  f 01 +  1 x − A 1 ω − B A B A   = 1 x  f 01 +  1 x − A  ω ω − B A − 1    = 1 x ( f 01 + ω M − W ) . Com bining the tw o previous equations, we obtain: x ( R + cY M ) = − b f ( f 01 + ω M − W ) , whic h yields ( 16 ). F rom equations ( 15 ) and ( 16 ), w e deduce the following: Prop osition 4.2. L et Q ( x, y ) := − b f c  V ′ + cy − f 01 − xR b f  e V ′ + cx − f 10 − y e R b f  + 1 c 2 ( b f ω + cxy )  c 2 f 00 − b P + R e R b f  . (17) Then Q ( x, Y ( x )) = 0 . Pr o of. Substituting y by Y ( x ) = − V ′ ( x ) + O ( x − 1 ) in ( 15 ), the left hand side v anishes, so that the righ t hand side is equal to zero as well, giving one expression for M : M = c 2 xf 00 − x b P + e R ( V ′ + cy − f 01 ) cx ( e V ′ + cx − f 10 ) + e Rω | y = Y ( x ) . Equation ( 16 ) gives a second expression for M : M = − xR + b f (+ V ′ + cY − f 01 ) b f ω + cxY . Therefore, the following polynomial in x and y  c 2 xf 00 − x b P + e R ( V ′ + cy − f 01 )   b f ω + cxy  −  cx ( e V ′ + cx − f 01 ) + e Rω   − xR + b f ( V ′ + cy − f 01 )  v anishes for y = Y ( x ) . Dividing it b y xc 2 , we get: 1 c 2  − c b f ( V ′ + cy − f 01 )( e V ′ + cx − f 10 ) + cxR ( e V ′ + cx − f 01 ) + cy e R ( V ′ + cy − f 01 ) + R e Rω + ( b f ω + cxy )( c 2 f 00 − b P )  , whic h is equal to Q ( x, y ) . 15 4.2. Algeb raicit y of b f In this section, w e establish a strategy to derive an algebraic equation on b f ( ω ) from Prop osition 4.2 , and compare it with the kernel method that is detailed in App endix A . Prop osition 4.3. W e have Q ( x, y ) = E ( x, y ) . (18) Pr o of. Both Q ( x, y ) and E ( x, y ) are p olynomials in Q ( c )[ t i , e t j , ω , f ij ][[ t ]][ x, y ] , and b oth of them hav e degree d in x and e d in y . Moreo v er, b oth of them v anish when y is substituted by Y ( x ) (and likewise when x is substituted b y X ( y ) , by symmetry). Since E is irreducible (this is clear from the fact that the asso ciated algebraic curve { ( x ( z ) , y ( z )) | z ∈ CP 1 } is connected), necessarily this implies that Q is equal to E up to a multiplicativ e factor in Q ( c, t i , e t j , ω , f ij )(( t )) . T o identify this prop ortionalit y constan t, let us lo ok at the highest degree co efficient in y in b oth p olynomials: • the highest degree monomial in y in E comes from the term cy e V ′ and is equal to − c e t e d y ˜ d . • The highest degree monomial in y in Q comes from the terms − b f y e V ′ + e Ry 2 , and is equal to b f e t e d y e d − f 00 e t e d y e d = − c e t e d y ˜ d as well. Using Prop osition 4.3 , we can pro ve our general algebraicit y result: Theorem 4.4. The c o efficients of monomials in x and y of the p olynomial Q ( x, y ) − E ( x, y ) induc e a system of ( d − 1)( e d − 1) p olynomials C i,j ∈ Q ( c )[ t i , e t j , ω ][[ t ]][ g 0 , 0 , . . . , g d − 2 , 0 , g 1 , 1 , . . . , g d − 2 , e d − 2 ] , 0 ≤ i ≤ d − 2 , 0 ≤ j ≤ e d − 2 , that satisfy C i,j ( f 0 , 0 , . . . , f d − 2 , 0 , f 1 , 1 , . . . , f d − 2 , e d − 2 ) = 0 for al l i, j , and whose Jac obian determinant det J = det( ∂ C i,j ∂ g kℓ ) evaluate d at ( f 0 , 0 , . . . , f d − 2 , e d − 2 ) is a non-zer o element of Q ( c )[ t i , e t j , ω ] . Conse quently, al l the series f ij , and in p articular b f = f 00 − c , ar e algebr aic over Q ( c, t i , e t j , ω )(( t )) . Pr o of. Note that the co efficien ts in y e d − 1 in E and Q are also trivially equal, as b oth are equal to − c e t e d − 2 − e t e d − 1 V ′ ( x ) (and lik ewise for x d − 1 ). Thus, w e ha ve indeed ( d − 1)( e d − 1) p olynomial relations: [ x i y j ]( Q − E ) , 0 ≤ i ≤ d − 2 , 0 ≤ j ≤ e d − 2 , that we denote b y C i,j . Let us also write Q i,j for [ x i y j ] Q ( x, y ) . W e first expand R and b P in x and y : R = − X 0 ≤ i ≤ d − 2 x i X i +2 ≤ k ≤ d t k f k − 2 − i, 0 , b P = X 0 ≤ i ≤ d − 2 0 ≤ j ≤ e d − 2 x i y j X i +2 ≤ k ≤ d j +2 ≤ ℓ ≤ e d t k e t ℓ f k − 2 − i,ℓ − 2 − j . Hence, for 1 ≤ i ≤ d − 2 , 1 ≤ j ≤ e d − 2 , ( i, j )  = ( d − 2 , e d − 2) , (1 , 1) , Q i,j = − b f c t i +1 e t j +1 + 1 c 2 X i +2 ≤ k ≤ d j +2 ≤ ℓ ≤ e d t k e t ℓ  ω f k − 2 − i, 0 f 0 ,ℓ − 2 − j − ω b f f k − 2 − i,ℓ − 2 − j − cf k − 1 − i,ℓ − 1 − j  , and, for 1 ≤ i ≤ d − 2 , Q i, 0 = b f c t i +1 f 1 , 0 − ct i + 1 c X i +1 ≤ k ≤ d t k ( f 1 , 0 f k − i − 1 , 0 − cf k − i, 0 )+ ω c 2 X i +2 ≤ k ≤ d 2 ≤ ℓ ≤ e d t k e t ℓ  f k − 2 − i, 0 f 0 ,ℓ − 2 − b f f k − 2 − i,ℓ − 2  , 16 and finally Q d − 2 , e d − 2 = t d − 1 e t e d − 1 + ω c t d e t e d f 0 , 0 − 1 c t d e t e d f 1 , 1 , Q 1 , 1 = − b f c t 2 e t 2 + c 2 + 1 c 2 X 3 ≤ k ≤ d 3 ≤ ℓ ≤ e d t k e t ℓ  ω f k − 3 , 0 f 0 ,ℓ − 3 − ω b f f k − 3 ,ℓ − 3 − cf k − 2 ,ℓ − 2  , Q 0 , 0 = − b f c f 1 , 0 f 0 , 1 + b f ω f 0 , 0 + ω c 2 X 2 ≤ k ≤ d 2 ≤ ℓ ≤ e d t k e t ℓ  f k − 2 , 0 f 0 ,ℓ − 2 − b f f k − 2 ,ℓ − 2  . Note that from their definition, all the series f i,j with ( i, j )  = (0 , 0) are p o wer series in c − 1 without constan t term, while b f = − c + t/ω + g , where g is a p ow er series in c − 1 without constant term. Hence, (det J )( f i,j ) is a Laurent series in c with a finite upp er b ound on its p ow ers. T o sho w that it is non-zero, it therefore suffices to chec k that its highest degree co efficien t in c is non-zero. Let us now lo ok at dep endence in c of the elements of J . F or i, j, k , ℓ ≥ 1 , w e hav e (with the con ven tion that t m = 0 , t n = 0 if the indices m , n are not within the correct ranges): ∂ C i,j ∂ f k,ℓ = − 1 c 2 ω b f t k + i +2 e t ℓ + j +2 − 1 c t k + i +1 e t ℓ + j +1 = 1 c ( t k + i +2 e t ℓ + j +2 ω − t k + i +1 e t ℓ + j +1 ) + O ( c − 2 ) , ∂ C i,j ∂ f k, 0 = O ( c − 1 ) , ∂ C i, 0 ∂ f k, 0 = − t k + i (1 + δ ( i, 1) ) + O ( c − 1 ) , ∂ C 0 ,j ∂ f k, 0 = O ( c − 1 ) , ∂ C i,j ∂ f 0 , 0 = − 1 c t i +1 e t j +1 + O ( c − 2 ) , ∂ C i, 0 ∂ f 0 , 0 = O ( c − 2 ) , ∂ C 0 , 0 ∂ f 0 , 0 = − cω + O (1) . Th us, the contribution to (det J )( f i,j ) corresp onding to picking the elemen ts ∂ C d − i − 1 , e d − j − 1 ∂ f i,j for i, j  = 0 , ∂ C d − i − 1 , 0 ∂ f i, 0 for i ≥ 1 and likewise for the f 0 ,j , and ∂ C 0 , 0 ∂ f 0 , 0 , is equal to ( − 1) d e d + d + e d +1 ( t d ) ( d − 2)( e d − 1) ( e t e d ) ( d − 1)( e d − 2) c 1 − ( d − 2)( e d − 2) + O ( c − ( d − 2)( e d − 2) ) , and every other term in (det J )( f i,j ) is O ( c − ( d − 2)( e d − 2) ) . This concludes the pro of. Theorem 4.4 giv es a strategy to obtain an algebraic equation on b f ( ω ) that do es not rely on the k ernel metho d, contrary to the strategy starting from ( 16 ) detailed in App endix A . These tw o strategies differ conceptually , and they also entail different computational steps. Let us conclude this subsection b y comparing them on these tw o fronts. In terms of computational steps, the k ernel method (see Appendix A for d etails and notation) generically requires the elimination of 2 d auxiliary v ariables (the M i ’s and the ¯ Z i ), and in volv es (among other steps) the iden tification of the ¯ Z i ’s as the d formal p o wer series solutions of an equation of degree d + e d . Our strategy generically requires the elimination of ( d − 1)( e d − 1) − 1 auxiliary series (the f i,j with ( i, j )  = (0 , 0) ). It is straightforw ard to inductively eliminate the f i,j with i, j  = 0 using the equation C d − i − 1 ,d − j − 1 , and the resulting p olynomial system on the remaining auxiliary series f i, 0 , f 0 ,j is comprised of d + e d − 4 quadratic equations in as many v ariables. In full generality , it is not clear to sa y which strategy is the simplest. How ever, in the sp ecial case of Ising quadrangulations (see Section 5 ), we can use the parity of the p oten tials to simplify b oth strategies. With the strategy of Theorem 4.4 , w e only need to solv e a linear system on 3 v ariables ( f 22 , f 20 , f 02 ). Compared to this, the 17 k ernel metho d, while simpler than in the generic case, still requires solving a quartic degree equation on ( ¯ Z i ) 2 , which is muc h more complicated. Another computational p oin t of comparison is the amoun t of information required on the mono chro- matic generating function Y ( x ) . While we use the known explicit rational parametrization of ( x, Y ( x )) in Section 5 , in principle w e only need the explicit expression of E ( x, y ) and the initial co efficien ts D ij , while the k ernel method requires the whole rational parametrization, since it is necessary to be able to write an equation on M (or M ) that is p olynomial in the catalytic v ariable. In particular, our metho d migh t b e generalized to other problems where the equiv alent of Y ( x ) is known to b e algebraic, with an explicit annihilating p olynomial, but where there is no known explicit parametrization. On a more conceptual level, our strategy uses t wo catalytic v ariables x and y , corresp onding to tw o differen t non-alternating parts of the b oundary . This migh t seem more complicated to eliminate than the single catalytic v ariable of the k ernel metho d, since there is no general framework to eliminate sev eral catalytic v ariables from a functional equation. Ho wev er, these tw o catalytic v ariables are “au- tomatically eliminated” in a sense, when we turn to the co efficien ts in x and y of Q − E . This is p ossible in particular b ecause x, y do not only corresp ond to refinements of the enumeration problem asso ciated to b f ( ω ) , but hav e an indep endent combinatorial meaning in the en umeration of the mono chromatic b oundary condition. Another notable feature of our strategy is that it k eeps the x ↔ y symmetry of the problem, which is clearly not the case of the kernel metho d applied to M . 4.3. Re-derivation of the T utte equation and kernel system Recall from Section 2.2 that equation ( 10 ) (equation (27) in [BC21] ) is the starting p oin t of applying the kernel metho d to b f , as mentioned in [BC21] and detailed here in Appendix A . The follo wing result relates it to the splitting equations that we ha ve deriv ed: Prop osition 4.5. Equation ( 16 ) implies e quation ( 10 ) (e quation (27) in [BC21] ). Pr o of. W e can rewrite the righ t hand side R ( x, w ) of ( 10 ) using a splitting equation: R ( x, w ) = cw xW ( x ) Y ( x ) + w " xV ′ ( x ) M ( 1 w , x ) w # x ≥ 0 = cw xW ( x ) Y ( x ) +  xV ′ ( x ) − AV ′ ( A ) ( x − A )(1 /w − B A )  = cw xW ( x ) Y ( x ) + xR ( x, 1 /w ) −  AV ′ ( A ) − xV ′ ( A ) ( x − A )(1 /w − B A )  = cw xW ( x ) Y ( x ) + xR ( x, 1 /w ) +  V ′ ( A ) 1 /w − B A  = cw xW ( x ) Y ( x ) + xR ( x, 1 /w ) −  c  1 1 /w − B A B  −  1 1 /w − B A B  1 1 /w − B A   " ↑ Splitting of c  1 1 /w − B A B  # , i.e. R ( x, w ) = cw xW ( x ) Y ( x ) + xR ( x, 1 /w ) + b f (1 /w ) f 10 (1 /w ) . (19) Since equation ( 16 ), with ω = 1 /w , is equiv alent to − b f ( 1 w ) w − cxY ( x ) ! M  1 w , x  = xR  x, 1 w  − b f  1 w  W ( x ) + b f  1 w  f 01  1 w  , 18 it implies that  c − f 00  1 w  − cwxY ( x )  M ( 1 w , x ) w − W ( x ) ! = cw xW ( x ) Y ( x ) + xR  x, 1 w  + b f  1 w  f 01  1 w  , whic h giv es equation (27) in [BC21] by sp ecializing c to 1. Remark 4.1. Equations ( 16 ) and ( 10 ) ar e b oth T utte-like e quations of the same gener ating function M , but her e c omp ar e d to [BC21] we c onsider e d mor e gener al b oundary c onditions with extr a B ’s to b e able to write ( 16 ) . Let us also note that from our master equation ( 18 ), w e can also obtain a “symmetrized” version of the k ernel system { K = 0 , R = 0 } of [BC21] for the unique formal p ow er series w ( ξ ) satisfying the k ernel relation ( 11 ). Indeed, substituting ω by Ω( ξ ) = w ( ξ ) − 1 in Q , we obtain: Q ( x, y ) | ω =Ω( xy ) = − b f (Ω) c  V ′ ( x ) + cy − f 01 (Ω) − xR (Ω) b f  e V ′ ( y ) + cx − f 10 − y e R (Ω) b f (Ω)  = − 1 c b f (Ω)  − cxy V ′ Ω − c 2 xy 2 Ω − f 01 (Ω) b f − xR (Ω)  · − cxy e V ′ Ω − c 2 x 2 y Ω − f 10 (Ω) b f − y e R (Ω) ! = − 1 c b f (Ω)  cxy w ( xy )( V ′ + cy ) + f 01 (Ω) b f + xR (Ω)  ·  cxy w ( xy )( e V ′ + cx ) + f 10 (Ω) b f + y e R (Ω)  . Therefore, p erforming the additional substitution x = x ( z ) , y = y ( z ) , we obtain, using ( 19 ) (with the sligh t abuse of notation w ( z ) = w ( x ( z ) y ( z )) , and after sp ecialization of c to 1, to fit with [BC21] ): 0 = Q ( x ( z ) , y ( z )) | ω =Ω( z ) = − 1 c b f (Ω) R ( x ( z ) , w ( z )) · e R ( y ( z ) , w ( z )) , so that w ( z ) must satisfy either R ( x ( z ) , w ( z )) = 0 or b R ( y ( z ) , w ( z )) = 0 . Note that this giv es tw o c hoices for the second equation of the system on ( w ( z ) , A ( w ( z )) in terms of z , but (in cases where w e do not need to eliminate z , e.g. bicolored m -angulations) these should lead to the same expression for the solution in terms of ξ = x ( z ) y ( z ) , as we kno w that w ( ξ ) is determined uniquely as a formal p ow er series in ξ − 1 . 5. Explicit solution fo r Ising quadrangulations In this section, w e tak e V ( x ) = − t 4 4 x 4 − t 2 2 x 2 and e V ( y ) = − e t 4 4 y 4 − e t 2 2 y 2 , which corresp onds to quadran- gulations decorated by the Ising mo del as explained in Section 3.2 , that we call Ising quadr angulations for short. W e apply to this choice of p oten tials the general strategy of Theorem 4.4 . 5.1. P a rametrization fo r the mono chromatic b oundary With that choice of V and e V , the parametrization of [Eyn16, Theorem 8.3.1] is written as x ( z ) = γ z + α 1 z − 1 + α 3 z − 3 y ( z ) = γ z − 1 + β 1 z + β 3 z 3 , (20) 19 and the system ( 6 ) characterizing the functions α i , β j , γ can b e written as (see the SageMa th com- panion file for details) − cβ 3 = γ 3 t 4 , − cα 3 = γ 3 e t 4 , − cβ 1 = 3 t 4 γ 2 α 1 + t 2 γ , − cα 1 = 3 e t 4 γ 2 β 1 + e t 2 γ , − 1 c = − γ 2 + α 1 β 1 + 3 α 3 β 3 . (21) These relations allo w us to express the original v ariables t, t 2 , t 4 , e t 2 , e t 4 , as rational functions of the parameters α 1 , α 3 , β 1 , β 3 , γ , and the original v ariable c : t = c ( γ 2 − α 1 β 1 − 3 α 3 β 3 ) , t 2 = c γ 2 ( β 1 γ − 3 α 1 β 3 ) , t 4 = cβ 3 γ 3 , e t 2 = c γ 2 ( α 1 γ − 3 β 1 α 3 ) , e t 4 = cα 3 γ 3 . (22) 5.2. Explicit master equation fo r b f W e apply the general strategy of Theorem 4.4 to Ising quadrangulations, and use the ab ov e parametriza- tion ( 22 ), to obtain the following: Prop osition 5.1. In the c ase of Ising quadr angulations, the p air ( ω , b f ( ω )) satisfies an explicit p olyno- mial e quation with c o efficients in Q [ c, γ , α 1 , α 3 , β 1 , β 3 ] , of de gr e e 3 in b f and 7 in ω , and the asso ciate d algebr aic curve is of genus 0. Mor e over, the p air of functions ζ ( ω ) = ω + γ 6 α 3 β 3 ω , λ ( ω ) = b f ( ω ) satisfies an explicit p olynomial e quation with c o efficients in Q [ c, γ , α 1 , α 3 , β 1 , β 3 ] , of de gr e e 3 in b oth ζ and λ . T o show Prop osition 5.1 , we first obtain an equation on ω , b f whose co efficien ts depend on the initial terms D ij of the Dobrushin b oundary condition: Lemma 5.2. The p air ( ω , b f ( ω )) satisfies: Eq ( b f , ω ) = 0 , 20 wher e Eq ( b f , ω ) = ( − t 3 4 e t 3 4 ω 7 + 2 c 2 t 2 4 e t 2 4 ω 5 − c 4 t 4 e t 4 ω 3 ) b f 3 + − ct 3 4 e t 3 4 w 7 + t e t 3 4 e t 3 4 w 6 + c ( c 2 − t 2 e t 2 ) t 2 4 e t 2 4 ω 5 − c 2 t 4 e t 4 ( e t 2 2 t 4 + t 2 2 e t 4 + 2 tt 4 e t 4 ) ω 4 + c 3 ( c 2 − t 2 e t 2 ) t 4 e t 4 ω 3 + c 4 tt 4 e t 4 ω 2 − c 7 ω ! b f 2 + − ( D 11 ct 3 4 e t 3 4 + ( c 2 + t 2 e t 2 ) c 2 t 2 4 e t 2 4 ) ω 5 + ( − c 2 ( t 4 e t 2 2 + e t 4 t 2 2 ) + D 20 t 2 t 4 e t 2 4 + D 02 e t 2 t 2 4 e t 4 + 2 tt 2 e t 2 t 4 e t 4 ) ct 4 e t 4 ω 4 + 2( tt 2 2 e t 4 + t e t 2 2 t 4 + D 02 e t 2 t 4 e t 4 + D 20 e t 2 t 4 e t 4 + cD 11 t 4 e t 4 − c 2 t 2 e t 2 t 4 e t 4 + c 4 ) c 2 t 4 e t 4 ω 3 + (2 tt 2 e t 2 t 4 e t 4 + D 02 t 2 t 4 e t 2 4 + D 20 e t 2 t 2 4 e t 4 − c 2 e t 2 2 t 4 − c 2 t 2 2 e t 4 ) c 3 ω 2 − ( c 3 + c t 2 e t 2 + D 11 t 4 e t 4 ) c 5 ω ! b f +( D 02 t 2 e t 4 + D 20 e t 2 t 4 + D 22 t 4 e t 4 + tt 2 e t 2 − c 2 t ) c 2 t 2 4 e t 2 4 ω 4 +  D 02 c 2 e t 2 t 4 e t 4 + D 20 c 2 t 2 t 4 e t 4 − D 02 tt 2 t 4 e t 2 4 − D 20 t e t 2 t 2 4 e t 4 − D 02 D 20 t 4 e t 4 + c 2 tt 2 2 e t 4 + c 2 t e t 2 2 t 4 − t 2 t 2 e t 2 t 4 e t 4 − c 4 t 2 e t 2  ct 4 e t 4 ω 3 +  2 c 2 tt 2 e t 2 t 4 e t 4 + 2 c 4 tt 4 e t 4 − t 2 e t 2 2 t 2 4 e t 4 − t 2 t 2 2 t 4 e t 2 4 − c 4 e t 2 2 t 4 − c 4 t 2 e t 4 − D 2 02 t 2 4 e t 3 4 − D 2 20 t 3 4 e t 2 4 − 2 D 02 t e t 2 t 2 4 e t 2 4 − 2 D 20 tt 2 t 2 4 e t 2 4 − 2 D 22 c 2 t 2 4 e t 2 4  c 2 ω 2 +  c 2 t e t 2 2 t 4 + c 2 tt 2 2 e t 4 + c 2 D 02 e t 2 t 4 e t 4 + c 2 D 20 t 2 t 4 e t 4 − c 4 t 2 e t 2 − D 02 tt 2 t 4 e t 2 4 − D 20 t e t 2 t 2 4 e t 4 − D 02 D 20 t 2 4 e t 2 4 − t 2 t 2 e t 2 t 4 e t 4  c 3 ω + c 6 ( tt 2 e t 2 + D 02 t 2 e t 4 + D 20 e t 2 t 4 + D 22 t 4 e t 4 − c 2 t ) . (23) Pr o of. T o obtain ( 23 ), w e start from the system of equations [ x k y ℓ ] Q = [ x k y ℓ ] E , 0 ≤ k ≤ d − 2 , 0 ≤ ℓ ≤ e d − 2 , and eliminate the auxiliary functions f ij for ( i, j )  = (0 , 0) . Let us start b y writing do wn the explicit expressions of the functions R , e R, b P that app ear in Q , and P that app ears in E . W e ha v e R ( x ) = − t 4 ( x 2 f 00 + xf 10 + f 20 ) − t 2 f 00 , e R ( y ) = − e t 4 ( y 2 f 00 + y f 01 + f 02 ) − e t 2 f 00 . Note that f 11 = ⟨ 1 ω − B A B A ⟩ = ω f 00 − t (we could rederive this from the co efficient of x 2 y 2 in Q − E , but it is quick er to write it immediately). Thus w e hav e for b P : b P ( x, y ) =( t 4 x 2 + t 2 )( e t 4 y 2 + e t 2 ) f 00 + t 4 x ( e t 4 y 2 + e t 2 ) f 10 + e t 4 y ( t 4 x 2 + t 2 ) f 01 + t 4 e t 4 xy ( ω f 00 − t ) + t 4 ( e t 4 y 2 + e t 2 ) f 20 + e t 4 ( t 4 x 2 + t 2 ) f 02 + t 4 e t 4 xf 12 + t 4 e t 4 y f 21 + t 4 e t 4 f 22 . W e express P in terms of the functions D ij =  B i A j  similarly to b P (note that D 11 cannot immediately b e expressed in a simpler wa y , con trary to f 11 ): P ( x, y ) = x 2 y 2 t 4 e t 4 D 00 + x 2 y t 4 e t 4 D 10 + xy 2 t 4 e t 4 D 01 + xy t 4 e t 4 D 11 + x 2 t 4 ( e t 4 D 02 + e t 2 D 00 ) + y 2 e t 4 ( t 4 D 20 + t 2 D 00 ) + xt 4 ( e t 4 D 12 + e t 2 D 10 ) + y e t 4 ( t 4 D 21 + t 2 D 01 ) + ( t 4 e t 4 D 22 + t 4 e t 2 D 20 + t 2 e t 4 D 02 + t 2 e t 2 D 00 ) . W e then use the parity of the p otentials, whic h implies that D ij = f ij = 0 if i + j is o dd, so that R ( x ) = − x 2 t 4 f 00 − ( t 4 f 20 + t 2 f 00 ) , e R ( y ) = − y 2 e t 4 f 00 − ( e t 4 f 02 + e t 2 f 00 ) , 21 b P ( x, y ) = x 2 y 2 ( t 4 e t 4 f 00 ) + xy ( t 4 e t 4 ( ω f 00 − t )) + x 2 ( t 4 e t 2 f 00 + t 4 e t 4 f 02 ) + y 2 ( t 2 e t 4 f 00 + t 4 e t 4 f 20 ) + ( t 2 e t 2 f 00 + t 4 e t 2 f 20 + t 2 e t 4 f 02 + t 4 e t 4 f 22 ) , and P ( x, y ) = x 2 y 2 ( t 4 e t 4 D 00 ) + xy ( t 4 e t 4 D 11 ) + x 2 t 4 ( e t 4 D 02 + e t 2 D 00 ) + y 2 e t 4 ( t 4 D 20 + t 2 D 00 ) + ( t 4 e t 4 D 22 + t 4 e t 2 D 20 + t 2 e t 4 D 02 + t 2 e t 2 D 00 ) . W e then plug these in to the expressions for the resp ectiv e expressions ( 8 ) for E and ( 17 ) for Q . As de- tailed in the SageMa th companion file, we then consider the system of equations on { ω , b f , f 20 , f 02 , f 22 } induced by stating that the co efficien ts in x i y j of Q − E must b e zero. The equations corresp onding to the co efficien ts in x 2 , y 2 and xy allow us to eliminate f 20 , f 02 , f 22 . W riting do wn the constan t co efficien t in Q ( x, y ) − E ( x, y ) , w e are left with a p olynomial equation in b f and ω , with co efficien ts in Q [ c, t, t i , e t j , D 20 , D 02 , D 11 , D 22 ] : 0 = Eq ( b f , ω ) , where Eq ( b f , ω ) is given in the statement of the prop o- sition. Pr o of of Pr op osition 5.1 . In equation ( 23 ), the functions D ij are for the momen t non-explicit functions in ( c, t, t k , e t ℓ ) . W e now make use of the rational parametrization ( 20 ) and the expression ( 22 ) of ( t, t k , e t ℓ ) in terms of the parameters ( α m , β n , γ , c ) , to write the D ij themselv es as explicit rational functions (and actually , p olynomials) in ( α m , β n , γ , c ) , through a Lagrange inv ersion form ula. Indeed, from ( 20 ), denoting ¯ x := x − 1 and ¯ z := z − 1 , we ha ve ¯ z = ¯ x  γ + α 1 ¯ z 2 + α 3 ¯ z 4  =: ¯ x Φ( ¯ z ) y ( z ) = γ ¯ z + β 1 ¯ z − 1 + β 3 ¯ z − 3 =: Υ( ¯ z ) , (24) so that, by the Lagrange–Bürmann form ula, for any non-negativ e k , the co efficient of x − ( k +1) in Y ( x ) is [ ¯ x k +1 ] Y ( x ) = 1 k + 1 [ u k ]  Υ ′ ( u )Φ( u ) k +1  . (25) F rom this, w e directly obtain expressions for D 20 = [ x − 3 ] Y ( x ) and (b y symmetry) D 20 : D 20 = c  α 1 γ 3 − α 3 β 1 γ 2 − 6 α 1 α 3 β 3 γ − α 2 1 β 1 γ − α 3 1 β 3  , D 02 = c  β 1 γ 3 − β 3 α 1 γ 2 − 6 β 1 β 3 α 3 γ − β 2 1 α 1 γ − β 3 1 α 3  . (26) F or D 11 and D 22 , we mak e use of the splitting pro cedure to express them as com binations of functions D i 0 : D 11 =  AB  = 1 c  t 2 + e t 2 D 20 + e t 4 D 40  , D 22 =  A 2 B 2  = 1 c 2  t 3 + e t 4 D 2 20 + 3 t e t 2 D 20 + ( e t 2 2 + 3 t e t 4 ) D 40 + 2 e t 2 e t 4 D 60 + e t 2 4 D 80  , so that, applying ( 25 ), we get: D 11 = c  γ 4 − α 1 β 1 γ 2 − 5 α 3 β 3 γ 2 − ( α 2 1 β 3 + β 2 1 α 3 ) γ + 3 α 2 3 β 2 3 − α 1 β 1 α 3 β 3  , D 22 = c  γ 6 − 6 α 3 β 3 γ 4 − 2( α 2 1 β 3 + β 2 1 α 3 ) γ 3 + (4 α 2 3 β 2 3 − 11 α 1 β 1 α 3 β 3 − α 2 1 β 2 1 ) γ 2 − (2 α 2 1 α 3 β 2 3 + 2 β 2 1 α 2 3 β 3 + α 3 1 β 1 β 3 + α 1 β 3 1 α 3 ) γ − α 2 1 β 2 1 α 3 β 3 + 2 α 1 β 1 α 2 3 β 2 3 − α 3 3 β 3 3  . (27) Injecting ( 22 ), ( 26 ) and ( 27 ) into ( 23 ), we get an explicit p olynomial equation Eq 2 ( ω , b f ) of degree 7 in ω and 3 in b f , with co efficients that are now in Q [ c, α 1 , β 1 , α 3 , β 3 , γ ] . As it is a bit long, and 22 straigh tforward to obtain from ( 23 ) as explained, w e do not write it here, but it is av ailable in the Sa geMa th companion file. The p olynomial Eq 2 ( ω , b f ) greatly simplifies when w e p erform the following transformation:  ω , b f  7→  ζ = ω + γ 6 α 3 β 3 ω , λ = b f ω  . (28) W e first straigh tforw ardly con vert the equation from ( ω , b f ) to ( ω , λ = b f ω ) , by taking the n umerator Eq 3 ( ω , λ ) of Eq 2 ( ω , λ ω ) . This already decreases the degree in ω from 7 to 6. F or ζ , we take the resultant of Eq 3 ( ω , λ ) and D = α 3 β 3 ω 2 − α 3 β 3 ζ ω + γ 6 . This resultan t is actually the square of a p olynomial in ζ and λ with co efficients in Q [ c, α 1 , β 1 , α 3 , β 3 , γ ] , that w e take to b e our equation Eq 4 ( ζ , λ ) for these new v ariables. It has degree 3 in b oth ζ and λ . W e hav e c heck ed using Maple that Eq 2 ( ω , b f ) = 0 is of genus 0 (and a fortiori so is Eq 4 ( ζ , λ ) ) 1 . Ho wev er, ev en for the simpler p olynomial Eq 4 ( ζ , λ ) , it is not possible to obtain immediately a rational parametrization with a command such as algcurves[parametrization] in Maple , due to the pres- ence of man y (six) parameters. It should b e p ossible to obtain an explicit rational parametrization in a “bruteforce” manner using in terp olation ov er the parameters. W e set aside this question for now, and fo cus on the sp ecial case of symmetric Ising quadrangulations (corresp onding to the sp ecialization t i = e t i ), where we obtain an explicit rational parametrization of ( ω , b f ) with minimal use of computer algebra. 5.3. The symmetric case W e now deriv e the explicit rational parametrization for the case of symmetric Ising quadrangulations, announced in Theorem 1.2 Pr o of of The or em 1.2 . All our parameters our now symmetric: α 3 = β 3 , α 1 = β 1 , t 4 = e t 4 = cα 3 γ 3 , t 2 = e t 2 = cα 1 γ − 3 α 3 γ 2 , t = c ( γ 2 − α 2 1 − 3 α 2 3 ) , so that the D ij are equal to D 00 = c ( γ 2 − α 2 1 − 3 α 2 3 ) = t, D 02 = D 20 = c ( γ 3 α 1 − γ 2 α 1 α 3 − γ α 3 1 − 6 γ α 2 3 α 1 − α 3 α 3 1 ) , D 22 = c ( γ 6 − 6 γ 4 α 2 3 − 4 γ 3 α 2 1 α 3 + γ 2 (4 α 4 3 − α 4 1 − 11 α 2 1 α 2 3 ) − γ ( α 4 1 α 3 + 4 α 2 1 α 3 3 ) − α 4 1 α 2 3 + 2 α 2 1 α 4 3 − α 6 3 ) . As detailed in the SageMa th companion file, the p olynomial Eq 4 , sym ( ζ , λ ) then factorizes into Eq 4 , sym ( ζ , λ ) = α 3 (2 γ 3 + α 3 ζ ) Eq sym ( ζ , λ ) , 1 W e used Maple for this b ecause the genus command of SageMa th is only implemented for curves ov er num ber fields. 23 where Eq sym ( ζ , λ ) ∈ Q [ c, α 1 , α 3 , γ ][ ζ , λ ] is equal to: Eq sym ( ζ , λ ) = ζ 2 cα 2 3 λ 2 + ζ  α 4 3 λ 3 + ( α 2 1 α 4 3 + 3 α 6 3 − α 4 3 γ 2 − 2 α 3 3 γ 3 ) cλ 2 +(3 α 8 3 − α 2 1 α 6 3 − 2 α 2 1 α 5 3 γ + 8 α 2 1 α 4 3 γ 2 − 5 α 6 3 γ 2 − 6 α 2 1 α 3 3 γ + α 2 1 α 3 3 γ 4 + α 4 3 γ 4 + α 2 3 γ 6 ) c 2 λ +( α 4 1 α 6 3 − 2 α 2 1 α 8 3 + α 1 3 − 4 α 4 1 α 5 3 γ + 4 α 2 1 α 7 3 γ + 6 α 4 1 α 4 3 γ 2 + 2 α 2 1 α 6 3 γ 2 − 4 α 8 3 γ 2 − 4 α 4 1 α 3 3 γ 3 − 8 α 2 1 α 5 3 γ 3 + α 4 1 α 2 3 γ 4 + 2 α 2 1 α 4 3 γ 4 + 6 α 6 3 γ 4 + 4 α 2 1 α 3 3 γ 5 − 2 α 2 1 α 2 3 γ 6 − 4 α 4 3 γ 6 + α 2 3 γ 8 ) c 3  − 2 α 3 γ 3 λ 3 + (9 α 2 1 α 4 3 γ 2 − 8 α 2 1 α 3 3 γ 3 − 6 α 5 3 γ 3 + α 2 1 α 2 3 γ 4 + 2 α 3 3 γ 5 ) cλ 2 +( − 6 α 4 1 α 5 3 γ + 14 α 4 1 α 4 3 γ 2 + 18 α 2 1 α 6 3 γ 2 − 10 α 4 1 α 3 3 γ 3 − 28 α 2 1 α 5 3 γ 3 − 6 α 7 3 γ 3 + 2 α 4 1 α 2 3 γ 4 + 12 α 2 1 α 3 3 γ 5 + 10 α 5 3 γ 5 − 2 α 2 1 α 2 3 γ 6 − 2 α 3 3 γ 7 − 2 α 3 γ 9 ) c 2 λ +( α 6 1 α 6 3 − 4 α 6 1 α 5 3 γ − 6 α 4 1 α 7 3 γ + 6 α 6 1 α 4 3 γ 2 + 20 α 4 1 α 6 3 γ 2 + 9 α 2 1 α 8 3 γ 2 − 4 α 6 1 α 3 3 γ 3 − 20 α 4 1 α 5 3 γ 3 − 20 α 2 1 α 7 3 γ 3 − 2 α 9 3 γ 3 + α 6 1 α 2 3 γ 4 − 4 α 2 1 α 6 3 γ 4 + 10 α 4 1 α 3 3 γ 5 + 36 α 2 1 α 5 3 γ 5 + 8 α 7 3 γ 5 − 4 α 4 1 α 2 3 γ 6 − 18 α 2 1 α 4 3 γ 6 − 12 α 2 1 α 3 3 γ 7 − 12 α 5 3 γ 7 + 12 α 2 1 α 2 3 γ 8 − 4 α 2 1 α 3 γ 9 + 8 α 3 3 γ 9 + α 2 1 γ 10 − 2 α 3 γ 11 ) c 3 . (29) Since it is of degree 2 in ζ , it will corresp ond to a rational curv e if it satisfies the so-called “one-cut prop ert y”. In other words, we w ant to chec k whether, when considering ( 29 ) as an equation in ζ with co efficien ts dep ending on λ , the associated discriminant ∆( λ ) (which is a p olynomial in λ ), only has 2 simple zeros, and the rest of its zeros are of even degree. Indeed, this means that ∆( λ ) can be written as ∆( λ ) = F ( λ ) 2 ( λ − a )( λ − b ) , where F is a p olynomial in λ . Then, we automatically get a joint rational parametrization of λ and ζ satisfying ( 29 ), via Zhuk ovsky’s transformation: λ ( H ) := a + b 2 + a − b 4  H + 1 H  , since we then hav e (for a given choice of the square ro ot) p ( λ − a )( λ − b ) = a − b 4  H − 1 H  , so that the tw o solutions ζ ± of ( 29 ) are also rational functions of H . Of these t wo solutions, only one has an asymptotic b ehavior when H → ∞ that is compatible with the initial definitions of λ and ζ as Lauren t series in ω (that satisfy λ, ζ → ∞ , λ/ζ → − c as ω → ∞ ), so that there is no ambiguit y in writing an expression for ζ ( H ) . Note that, since there is a symmetry λ ( 1 H ) = λ ( H ) , ζ ± ( 1 H ) = ζ ∓ ( H ) , the choice of lo oking at the behavior as H → ∞ and not H → 0 does not ultimately impact the form ulae for λ and ζ . These verifications and computations are straightforw ard to do with a computer algebra system (see the Sa geMa th companion file), and w e obtain the following rational parametrization of the algebraic curv e asso ciated to Eq sym ( ζ , λ ) : ζ sym ( H ) = 1 ( H α 1 + α 3 + γ ) 2 α 2 3  H 3 ( α 3 1 α 3 3 + α 3 1 α 2 3 γ ) + 2 H 2 γ α 2 1 α 2 3 (3 α 3 + 2 γ ) + H α 1 γ 2 (9 α 3 3 + 7 α 2 3 γ − α 3 γ 2 + γ 3 ) + α 3 γ 3 (2 α 2 3 + 4 α 3 γ + 2 γ 2 )  λ sym ( H ) = − c H ( H α 1 + α 3 + γ ) ( H ( α 3 + γ ) + α 1 ) . (30) 24 T o get to a parametrization for our initial v ariables ( ω , b f ) , note that the equation on ( H , ω ) given by ζ ( ω ) = ζ sym ( H ) is itself of degree 2 in ω , and w e chec k that it also satisfies the one-cut prop erty (see the Sa geMa th companion file for details), so that we find a joint parametrization ( H ( h ) , ω ( h )) for H and ω satisfying this relation given by ζ . Since b f is a rational function of ω and λ , and λ ( H ( h )) is also a rational function of h , this yields a joint rational parametrization ( ω ( h ) , b f ( h )) . As b efore, there is ultimately no imp ortance in the choice of a solution to the quadratic equation in ω , since the c hange h → 1 /h leads to ω ± → ω ∓ and leav es λ unchanged. W e thus obtain the announced rational parametrization of ( ω , b f ) : ω sym ( h ) = ( α 3 h + γ ) 2 γ 3 h ( γ h + α 3 ) 2 α 3 b f sym ( h ) = − c ( α 3 γ h 2 + h ( α 2 1 − 2 α 3 γ ) + α 3 γ )( γ h + α 3 ) 3 ( α 3 h + γ ) γ 4 ( h − 1) 2 h 2 . Remark 5.1. One c an che ck that sp e cializing α 1 = 0 in ( 4 ) gives b ack the p ar ametrization of [BC21] for pur e bic olor e d quadr angulations, with h = z 4 = s . As explained in Section 2.2 , a remark able feature of the parametrization obtained for ( ω , b f ) in the case of m -constellations in [BC21] , is that ω and b f are written as rational functions of x, Y ( x ) , and inherit from this a parametrization in z : thus, x, Y ( x ) and ω , b f ( ω ) can b e written jointly as rational functions of a single v ariable z , while satisfying the kernel r elation b f ω + cxY ( x ) = 0 . Here, in the case of Ising quadrangulations, we could not obtain a parametrization of ( ω , b f ) so directly . F urthermore, the explicit solution of this new sp ecial case allows us to pro ve Corollary 1.3 : Cor ol lary 1.3 . It obviously suffices to prov e that the prop erty do es not hold for the case of symmet- ric Ising quadrangulations. Substituting ω sym ( h ) , b f sym ( h ) , x ( z ) , y ( z ) in to the kernel relation gives a p olynomial equation on h and z . W e sho w that the asso ciated algebraic curv e is non-rational. W e actually obtain something ev en stronger. Consider the algebraic equation on H and Z given by λ sym ( H ) = − cξ ( Z ) , with ξ ( Z ) = x ( z ) y ( z ) | Z = z 2 (whic h is a rational function of Z ). It can b e written as cZ 2 ( H α 1 + α 3 + γ ) ( H ( α 3 + γ ) + α 1 ) = H  γ Z 2 + α 1 Z + α 3  ( γ + α 1 Z + α 3 Z ) . In particular, it is a polynomial of degree 2 in H , whose discriminant has 4 distinct simple p oles in Z (see the Sa geMa th companion file for details). Therefore, it cannot b e of gen us 0. A for- tiori , the relation λ ( H ( h )) = − cξ ( z 2 ) cannot corresp ond to a genus 0 curve. Since b oth rational parametrizations ( x ( z ) , y ( z )) and ( ω sym ( h ) , b f sym ( h )) are prop er, this implies that the tw o curv es ( ω , b f ) , ( x, Y ( x )) do not admit a joint rational parametrization of the form (with some abuse of notation) ( x ( S ) , y ( S ) , ω ( S ) , b f ( S )) satisfying the relation ω ( S ) b f ( S ) = − cx ( S ) y ( S ) . The question of lo oking for a parametrization of ω , b f as rational functions of x, y (i.e. “constructing ( ω , b f ) on the curve E ( x, y ) ”) without requiring the k ernel relation is left op en, and would b e related to broader questions on the genus of v ariable-separated curves (see for instance [Pak18] ). Ho wev er, w e b eliev e that the k ernel relation is a combinatorially significant relation. In particular, it also pla ys an imp ortan t role in the slice decomp osition of the alternating b oundary condition [CEL] . 25 6. Conclusion and p ersp ectives In this pap er, we hav e established a new strategy to obtain an explicit algebraic master equation on the generating functions of general h yp ermaps with an alternating b oundary , that prov es to b e muc h more efficient than the classical kernel metho d, at least in the case of Ising quadrangulations. This represen ts significant progress in the study of the alternating condition. There are still many op en questions regarding this combinatorial problem. First, we do not y et hav e an explicit formula for the master equation for general p otentials. Giv en the concrete examples that we hav e at hand, it is natural to conjecture that the asso ciated curve is alw ays of genus 0. If it is true, it w ould b e satisfying to hav e a general expression for a rational parametrization, like in the mono chromatic case. As men tioned in Section 2.2 , w e also exp ect the generating functions for general topologies to satisfy the top ological recursion. If it is indeed the case, it would be interesting to understand in particular how the nice relation b etw een the monochromatic and alternating conditions for m -constellations ev olve in higher top ologies, and if there is any relation b et ween the tw o sp ectral curv es in the general case. As a related question, it would b e interesting to hav e a b etter understanding of the k ernel relation, b ey ond its computational imp ortance. Another computational step that we would like to make sense of more conceptually is the operation of taking the co efficients in x and y of the equation Q = E , as this is a crucial step in our strategy . It migh t also give some insigh t on the relation b etw een the mono c hromatic and alternating conditions. Finally , to further the com binatorial understanding of h yp ermaps, it is also natural to w an t to obtain b oth the kno wn and exp ected formulae by bijective argumen ts. W e will address this in future w ork [CEL] , using the slice decomp osition metho d. A ckno wledgements This w ork was initially supported b y the ERC-SyG pro ject, Recursiv e and Exact New Quantum Theory (ReNewQuantum) which receiv ed funding from the Europ ean Research Council (ER C) under the Europ ean Union’s Horizon 2020 research and inno v ation programme under grant agreemen t No 810573. In particular this pro ject was b orn during the problem session of the Otranto summer school organized b y ReNewQuan tum. AC is currently supported by the Austrian Science F und (FWF) grant 10.55776/F1002. The authors thank Jérémie Bouttier and Thomas Lejeune for insight and v aluable discussions. A. Application of the k ernel metho d to general p otentials In this app endix, w e will detail ho w to write the T utte equation ( 10 ) obtained in [BC21] , in a wa y that fits in the following general theorem: Theorem A.1 (Theorem 3 in [BMJ06] ) . L et Q ( y 0 , y 1 , . . . , y k , t, v ) b e a p olynomial in k + 3 indetermi- nates, with c o efficients in a field K . W e c onsider the functional e quation F ( u ) ≡ F ( t, u ) = F 0 ( u ) + tQ ( F ( u ) , ∆ F ( u ) , ∆ (2) F ( u ) , . . . , ∆ ( k ) F ( u ) , t, u ) , (31) wher e F 0 ( u ) ∈ K [ u ] is given explicitly and the op er ator ∆ is the divide d differ enc e: ∆ F ( u ) = F ( u ) − F (0) u , 26 and ∆ ( i ) is its i -th iter ate. Then ( 31 ) has a unique solution F ( t, u ) ∈ K [ u ][[ t ]] , and this formal p ower series is algebr aic over K ( t, u ) . Here, u is the catalytic v ariable of the equation, whic h is the role play ed b y x in ( 10 ), and the main v ariable ( t in the theorem) w ould b e w . The field K would b e Q ( t, t i , e t j ) . The first obstruction in applying this theorem to ( 10 ) is that it contains not only p olynomial (or rational) functions in x , but also the function Y ( x ) , which is a formal p o wer series. Ho wev er, we can mak e use of the rational parametrization ( 5 ), to switc h from x to ¯ z = z − 1 as a catalytic v ariable. Then ( 10 ) writes, for M ( ¯ z , w ) = c M ( x ( z ) , w ) : M ( ¯ z , w ) = M ( ¯ z , w )( A ( w ) + w x ( z ) y ( z )) + w x ( z )( V ′ ( x ( z )) + y ( z )) y ( z ) − w X 2 ≤ i ≤ d t i h x ( z ) i ( M ( ¯ z , w ) + V ′ ( x ( z )) + Y ( x ( z ))) i x ≥ 0 . (32) Note that M 0 ( ¯ z ) = M ( ¯ z , 0) = 0 is clearly explicit, so that ( 32 ) can b e written as M ( ¯ z , w ) = M 0 ( ¯ z ) + w  A ( w ) w + x ( z ) y ( z )) M ( ¯ z , w ) + ( x ( z ) y ( z ))( V ′ ( x ) + y ( z )) + X 2 ≤ i ≤ d t i h x ( z ) i ( M ( ¯ z , w ) + V ′ ( x ( z )) + Y ( x ( z ))) i x ≥ 0  . Ev erything in factor of w in the righ t-hand side is an explicit rational function in ¯ z and M ( ¯ z , w ) , except A ( w ) /w = [ ¯ x ]( M ( x, w )) + t , and X 2 ≤ i ≤ d t i h x ( z ) i M ( ¯ z , w ) i x ≥ 0 whic h are explicit combinations of the co efficien ts of ¯ x, ¯ x 2 , . . . , ¯ x d , in the p o wer series c M ( x, w ) =: M ( ¯ x ) ∈ Q [ c, t, t i , e t j ][[ ¯ x, w ]] . They are first easily expressed in terms of the divided differences in ¯ x . Indeed, we ha ve ∆ ( i ) ¯ x M = M ( ¯ x ) − M (0) − ¯ xM ′ (0) − · · · − ¯ x i − 1 / ( i − 1)! M i − 1 (0) ¯ x i , so that [ ¯ x i ] M = 1 i ! M ( i ) (0) = ∆ ( i ) ¯ x M − ¯ x ∆ ( i +1) ¯ x M . No w, we claim that the divided differences in ¯ x can b e written in terms of the ones in ¯ z , in the following w ay , for any i ≥ 1 : ∆ ( i ) ¯ x M ( ¯ x ( ¯ z )) = Φ( ¯ z )   X 1 ≤ j ≤ i p i,j ( ¯ z )∆ ( j ) ¯ z M   , (33) where p i,j ( ¯ z ) are explicit p olynomials in Q [ c, γ , α k , β l ][ ¯ z ] . Indeed, as ¯ x ( ¯ z ) = ¯ z / Φ( ¯ z ) , where Φ( ¯ z ) = γ + P 0 ≤ k ≤ e d − 1 α k ¯ z k , we ha ve ∆ ¯ x M ( ¯ x ( ¯ z )) = M ( ¯ x ( ¯ z )) − M (0) ¯ x ( ¯ z ) = Φ( ¯ z ) · M ( ¯ z ) − M ( ¯ z = 0) ¯ z = Φ( ¯ z )∆ ¯ z M , since ¯ x ( ¯ z = 0) = 0 . The statemen t for general i follows b y a straigh tforward induction, applying the Leibniz rule for divided differences: ∆ x ( f ( x ) g ( x )) = f (0) · ∆ x g ( x ) + ∆ x f ( x ) · g ( x ) . 27 Equation ( 33 ) implies that the co efficient [ ¯ x i ] M is itself an explicit polynomial in the v ariables ¯ z , M , ∆ ¯ z M , . . . ∆ ( i +1) ¯ z M with co efficien ts in Q [ c, γ , α k , β ℓ ] , so that (up to m ultiplication by a pow er of ¯ z ), ( 32 ) can indeed by written in the form of ( 31 ), and Theorem A.1 applies. Let us end this section with a sk etch of the strategy of the kernel metho d , applied to general p otentials V , e V , following the exp osition given in [BMJ06, Section 3.1] . W e compare it with our “ Q = E ” metho d in Section 4.2 . The k ernel metho d consists in lo oking for formal p o wer series ¯ Z = ¯ Z ( ω ) ∈ Q [ c, γ , α k , β ℓ ][ ω ] that satisfy K ( ¯ Z , ω ) = 0 , (34) where K ( ¯ z , ω ) = K ( x ( z ) , ω ) . Indeed, for suc h a solution, equation ( 32 ) implies 0 = K ( ¯ Z i , w ) M ( ¯ Z i , w ) = R ( ¯ Z i , M 1 , . . . , M d , w ) , with M i = [ ¯ x i ] M ( ¯ x ) and R ( ¯ z , M 1 , . . . , M d , w ) = b R ( x ( z ) , w ) , where the dep endence on the M i is now emphasized. If w e ha v e d suc h solutions ¯ Z 1 , . . . , ¯ Z d , counted with multiplicit y , then this giv es a system of d equations on the d unkno wns M 1 , . . . , M d (if a solution ¯ Z i has non-trivial multiplicit y , one needs to write the equations asso ciated to the deriv ativ es of R at ¯ Z i ). W e can then solve this system and insert the solutions in ( 32 ), yielding an explicit p olynomial equation on M ( ¯ z , w ) . Theorem A.1 ensures that there is a unique solution to the p olynomial system on M 1 , . . . , M d , as these then determine M ( ¯ z , w ) . It remains to chec k that we ha v e d solutions to ( 34 ). Since we ha v e: x ( z ) y ( z ) = γ z + X 0 ≤ k ≤ e d − 1 α k z − k ! γ z − 1 + X 0 ≤ ℓ ≤ d − 1 β ℓ z ℓ ! = γ + X 0 ≤ k ≤ e d − 1 α k z − k − 1 ! γ + X 0 ≤ ℓ ≤ d − 1 β ℓ z ℓ +1 ! , to get to a p olynomial equation on ¯ z from K ( ¯ z , w ) , w e hav e to multiply it b y ¯ z d , yielding the follo wing p o wer series in w with p olynomial co efficients in ¯ z : Ψ( w , ¯ z ) = ¯ z d (1 − A ( w )) + w γ + X 0 ≤ k ≤ e d − 1 α k ¯ z k +1 ! γ ¯ z d + X 0 ≤ ℓ ≤ d − 1 β ℓ ¯ z d − ℓ − 1 ! . Since the constant co efficient in w is Ψ(0 , ¯ z ) = ¯ z d , [BMJ06, Theorem 2] implies that there are exactly d solutions to K ( x ( ¯ Z ) , w ) = 0 that are fractional p o wer series in w , counted with multiplicit y . 28 References [AB25] M. Alb enque and J. Bouttier. 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