Nearly Optimal Algorithms for the Decomposition of Multivariate Rational Functions and the Extended L"uroths Theorem

NEARL Y OPTIMAL ALGORITHMS FOR THE DECOMPOSITION OF MUL TIV ARIA TE RA TIONAL FUNCTIONS AND THE EXTENDED L ¨ UR OTH’S THE O REM GUILLA UME CH ` EZE Abstract. The extended L ¨ uroth’s Theorem says t hat if the transcendence de- gree of K ( f 1 , . . . , f m ) / K is 1 then there exists f ∈ K ( X ) such that K ( f 1 , . . . , f m ) is equal t o K ( f ). In this paper we show ho w to compute f with a probabilis tic algorithm. W e also describ e a probab ilis tic and a deterministic algorithm f or the decomposition of m ultiv ariate rational f unctions. The probabilistic algo- rithms prop osed i n this paper are softly optimal when n is fixed and d te nds to infinity . W e also giv e an indecomposabili ty test based on gcd computations and Newton’s pol ytope. In the last section, we show that we get a p olynomial time algori thm, with a minor mo dification in the exponent ial time dec omp o- sition algorithm pr oposed by Gutierez-Rubio-Sevilla in 2001. Introduction Polynomial deco mpo s ition is the problem of r epresenting a given p olynomial f ( x ) as a functional compos itio n g ( h ( x )) of p olynomials of smaller degree. This decomp osition has b een widely studied s ince 19 22, s ee [27], a nd efficien t algor ithms are kno wn in the univ a riate case, see [3, 9, 20, 37, 38] and in the mu ltiv ariate case [11, 37, 40]. The decomp osition of ra tional functions has a lso b een studied, [41, 1]. In the m ultiv ariate case the situatio n is the follo wing: Let f ( X 1 , . . . , X n ) = f 1 ( X 1 , . . . , X n ) /f 2 ( X 1 , . . . , X n ) ∈ K ( X 1 , . . . , X n ) be a ra tio- nal function, where K is a field and n ≥ 2. It is commonly sa id to be comp osite if it can b e written f = u ◦ h where h ( X 1 , . . . , X n ) ∈ K ( X 1 , . . . , X n ) and u ∈ K ( T ) such that deg ( u ) ≥ 2 (recall that the degree of a rational function is the maximum of the degr ees of its n umerator and denominator after reduction), o therwise f is said to be non-comp os ite. This decompo sition app ear s when we study the kernel of a deriv ation, see [24]. In [2 4] the author gives a multiv ariate rational function decomp ositio n algo rithm, but this alg orithm is not o ptima l a nd works only for fields of characteristic zer o . In this paper , we g ive a probabilis tic optimal a lgorithm. In other words, our al- gorithm decomp oses f ∈ K ( X 1 , . . . , X n ) with ˜ O ( d n ) arithmetic op erations , wher e d is the degree of f . W e supp ose in this w ork that d tends to infinity a nd n is fixed. W e use the classical O and ˜ O (“soft O ”) notation in the neighbor ho o d of infinit y a s defined in [3 9, Chapter 25.7 ]. Informally sp eaking , “ soft O ”s a re used for reada bilit y in order to hide logar ithmic factors in complexity estimates. Then, the size of the input and the num b er o f arithmetic ope r ations p erformed b y our algorithm hav e the same order of magnitude. This is the reas o n why we call our algorithm “optimal”. 1 2 G. CH ` EZE F urthermore , our algo rithm als o w orks if the characteristic of K is g reater than d ( d − 1) + 1. This decompo s ition a lso app ea r s when we study int ermediate fields o f an unira- tional field. In this situation, the pro blem is the following: we hav e m multiv a riate rational functions f 1 ( X ), . . . , f m ( X ) ∈ K ( X ), and we wan t to know if there exists a prop er intermediate field F s uch that K ( f 1 , . . . , f m ) ⊂ F ⊂ K ( X ). In the affirmative case, w e wan t to compute F . If tr.deg K ( F ) = 1 then by the extended L ¨ uroth’s Theorem, see [31, Theorem 3 p. 15] w e hav e F = K ( f ). Theorem 1 (Ex tended L ¨ uroth’s Theorem) . Le t F b e a field such t hat K ⊂ F ⊂ K ( X 1 , . . . , X n ) and tr.de g K ( F ) = 1 . Then ther e exists f ∈ K ( X 1 , . . . , X n ) such that F = K ( f ) . The cla ssical L ¨ uroth’s Theo rem is sta ted with univ a riate ra tional functions. The- orem 1 gives an extension to multiv ar iate ratio nal functions. This extended theor e m was fir st proved by Gordan in characteristic zero, see [13], and by Igusa in general, see [17]. T he r e e xist algorithms to compute f , called a L ¨ uroth’s generator, s ee e.g . [15, 25]. Thanks to the Extended L¨ uroth’s Theorem the computation of in ter mediate fields is divided into t wo parts: first w e compute a L¨ uroth’s generator f , and second we decomp ose f . Then f = u ◦ h , and F = K ( h ) is an intermediate field. In [15] the authors show that the decomp ositio n of f bijectively corres po nds to intermediate fields. They a lso give a lgorithms to co mpute a L ¨ uroth’s genera tor a nd to decomp ose it. Unfortunately , the decomp osition algo rithm has an e xpo nential time c o mplex- it y , but the complexity ana ly sis of this algorithm is to o pe ssimistic. Indee d, in the last sectio n of this pap er w e show that w e ca n mo dify it and get an algor ithm with a polyno mial time complexit y . The decomp ositio n of rational functions a lso app ears when we study the sp ec- trum of a rational function. In this paper w e use this p o int of view in order to g ive fast algorithms. Let K b e an algebraic closure of K . Let f = f 1 /f 2 ∈ K ( X 1 , . . . , X n ) b e a r ational function of degree d . The set σ ( f 1 , f 2 ) = { ( µ : λ ) ∈ P 1 K | µf 1 − λf 2 is reducible in K [ X 1 , . . . , X n ] , or deg ( µf 1 − λf 2 ) < d } is the sp ectrum of f = f 1 /f 2 . W e recall that a po lynomial reducible in K [ X 1 , . . . , X n ] is said to be abso lutely reducible. A classical theorem of Ber tini a nd K rull, see Theorem 22, implies that σ ( f 1 , f 2 ) is finite if f 1 /f 2 is non-comp osite. Actually , σ ( f 1 , f 2 ) is finite if and o nly if f 1 /f 2 is non-comp osite and if and only if the pencil of algebra ic curves µf 1 − λf 2 = 0 has an irr e ducible g eneral element (see for instance [18, Cha pitre 2, Th ´ eor` eme 3.4 .6 ] and [7, Theorem 2.2] for deta iled pro ofs). T o the a uthor’s knowledge, the first effective result a bo ut the sp ectrum has b een given b y Poincar´ e [2 6]. He show ed tha t | σ ( f 1 , f 2 ) | ≤ (2 d − 1 ) 2 + 2 d + 2. This b ound was improv ed b y Rupp ert [2 8] who proved that | σ ( f 1 , f 2 ) | ≤ d 2 − 1 . DECOMPOSITION OF M UL TIV ARIA TE RA TIONAL FUNCTIONS 3 This result was obtained as a bypro duct of a very in teresting techn ique de velop ed to decide the reducibilit y of an algebraic plane curv e. Several pap ers improve this result, see e.g. [23, 36, 2, 7, 4]. The previous result s ays that if f 1 /f 2 is a non- c o mpo site reduced r ational func- tion then for all but a finite n umber o f λ ∈ K we hav e: f 1 + λf 2 is absolutely irreducible (i.e. irr educible in K [ X 1 , . . . , X n ]). F urthermore, the num b er of “bad” v alues of λ is low er than d 2 − 1. Thu s we c a n deduce a pro babilistic test for the decomp osition of a rational function, base d o n a n absolute irreducibility test. In this paper we will give a deco mpo sition algor ithm bas e d o n this kind of idea. F urthermore , we will see that this algor ithm is softly o ptimal when the following hypotheses are satisfied: Hypo thesis (C): K is a perfect field of characteristic 0 o r at lea st d ( d − 1) + 1 . Hypo thesis (H): ( ( i ) deg( f 1 + Λ f 2 ) = deg X n ( f 1 + Λ f 2 ) , where Λ is a new v ariable , ( ii ) Res X n  f 1 (0 , X n ) + Λ f 2 (0 , X n ) , ∂ X n f 1 (0 , X n ) + Λ ∂ X n f 2 (0 , X n )  6 = 0 in K [Λ] . where deg X n f repres e nts the pa rtial deg ree of f in the v ariable X n , deg f is the total degree of f a nd Res X n denotes the resultant rela tively to the v aria ble X n . These hypotheses are necessary , b ecaus e w e will use the factorizatio n algo r ithms prop osed in [22], where these kinds o f h y p o thes es are needed. Actually , in [22] the author studies the fa c to rization of a p olyno mial F and uses h yp othesis (C) and hypothesis (L), where (L) is the follo wing: Hypo thesis (L): ( ( i ) deg X n F = deg F, and F is monic in X n , ( ii ) Res X n  F (0 , X n ) , ∂ F ∂ X n (0 , X n )  6 = 0 . If F is squarefree, then hypothes is (L) is no t restrictive since it can b e assured by means of a generic linear c ha nge of v ar ia bles, but w e will not discuss this question here (for a complete treatmen t in the biv ar iate case, see [10, Propo sition 1]). Roughly speaking , our h yp othesis (H) is the hypothesis (L) applied to the p oly- nomial f 1 + Λ f 2 . In (H , i ) we do no t as sume tha t f 1 + Λ f 2 is mo nic in X n . Indeed, after a generic linea r c ha nge of coo rdinates, the leading coefficient relatively to X n can b e written: a + Λ b , with a , b ∈ K . In o ur pro babilistic alg orithm, we ev aluate Λ to λ 6∈ σ ( f 1 , f 2 ), th us deg ( f 1 + λf 2 ) = deg( f 1 + Λ f 2 ) and a + λb 6 = 0. Then we can consider the monic part of f 1 + λf 2 and we get a polynomial sa tisfying (L, i ). Then (H, i ) is sufficien t in our situation. F urthermore, in this paper , we assume f 1 /f 2 to be reduced, i.e. f 1 and f 2 are coprime. W e re call in Lemma 6 that in this situatio n f 1 + Λ f 2 is squarefre e. Thus h y po thesis (H) is not restrictive. Complexity m o del. In this pap er the complexity estimates charge a constant cost for each arithmetic o p er ation (+, − , × , ÷ ) and the equality test. All the constants in the base fields (or rings) are though t to be freely at our disp o sal. In this pap er we supp ose that the numb er of variabl es n is fixe d a nd that the degree d tends to infinit y . F urthermor e, we sa y that a n algorithm is softly optimal if it w orks with ˜ O ( N ) arithmetic op erations where N is the size of the input. 4 G. CH ` EZE Polynomials are repres ent ed by dense vectors of their co efficie nts in the usual monomial basis. F or each in teger d , we ass ume that we ar e given a computation tree that computes the pro duct of t wo univ aria te p olyno mials of degr ee a t most d with at most ˜ O ( d ) op erations, indep endent ly of the base ring, see [39, Theorem 8.23]. W e use the co nstant ω to denote a fe asible matrix multiplic ation exp onent as defined in [39, Chapter 12]: tw o n × n matrices ov er K can be multiplied with O ( n ω ) field op erations. As in [8] we r e q uire that 2 < ω ≤ 2 . 3 76. W e rec a ll that the computation of a so lution basis of a linear sys tem with m equatio ns and d ≤ m unknowns o ver K tak es O ( md ω − 1 ) op e r ations in K [8, Chapter 2] (see a lso [33, Theorem 2.10]). In [22] the author gives a probabilistic (resp. deterministic) algor ithm for the m ultiv ar ia te r ational factorization. The rationa l facto r ization o f a polynomia l f is the factorization in K [ X ], where K is the co e fficie n t field of f . This alg orithm uses o ne factor ization of a univ a r iate p oly nomial o f degre e d and ˜ O ( d n ) (r esp. ˜ O ( d n + ω − 1 )) arithmetic op erations , where d is the total deg ree o f the polyno mial and n ≥ 3 is the num be r of v ariables. If n = 2, in [21],[22, E rrata], the author g ives a probabilis tic (r esp. deterministic) alg orithm for the ratio nal factoriza tion. The nu mber of arithmetic op eratio ns of this algor ithm b elongs to ˜ O ( d 3 ) (re s p. ˜ O ( d ω +1 )). W e note that for n ≥ 3 if the co st o f the univ ariate p olynomial facto rization b elong s to ˜ O ( d n ) then the probabilistic algorithm is softly optimal. Main Theorems . The following theorems g ive the complexity results ab out our algorithms. Althoug h we will us e no proba bilistic mo del of computation, we will informally s ay pr ob abilistic algorithms when spea king ab out the computatio n trees o ccurring in the next theorems. F or the s ake of precision, we pr efer to express the probabilistic asp ects in terms of families o f computation tr ees. Almo s t all the trees of a family a r e exp ected to b e e x ecutable on a g iven input (if the car dina lit y of K is large enough). Theorem 2. L et f = f 1 /f 2 b e a multivariate r ational function in K ( X 1 , . . . , X n ) of de gr e e d , t her e exists a family of c omputation t re es over K p ar ametrize d by z := ( a , b ) ∈ K 2 n such that: • Any exe cut able tr e e of the family r eturns a de c omp osition u ◦ h of f with h a non- c omp osite r ational function. • If a , b ar e not the r o ots o f some non-zer o p olynomials the tr e e c orr esp onding to z is exe cutable. F urt hermor e, we have: (1) An exe cutable tr e e p erforms two factoriza tions in K [ X 1 , . . . , X n ] of p olyno- mials with de gr e e d , and one c omputation of u . (2) Under hyp othesis (C) and (H) we have t his estimate: an ex e cutable tr e e p erforms one factorizatio n of a univariate p olynomial of de gr e e d over K plus a numb er of op er ations in K b elonging t o ˜ O ( d n ) if n ≥ 3 , or to ˜ O ( d 3 ) if n = 2 . Since we use the dense representation of f 1 and f 2 , the size of f is of the order of ma gnitude d n . The previo us statement thus ass erts that the complexity of our probabilistic algorithm is softly optimal for n ≥ 3. W e precise the condition “If a, b are not the ro o ts o f so me non-zero po lynomials” in Remark 13 and Rema rk 15. In characteris tic zero we can say that for almos t all z the tre e corres po nding to z DECOMPOSITION OF M UL TIV ARIA TE RA TIONAL FUNCTIONS 5 is executable. W e also give a deterministic decomp osition algo rithm. Theorem 3. If K is a field with a le ast max( d 2 , 3 2 d 2 − 2 d + 1 ) element s , then the de c omp osition f = u ◦ h , with h non- c omp osite, c an b e c ompute d with at m ost O ( d 2 ) absolute factorizations of p olynomials with de gr e e d , and at most O ( d 2 ) c omputa- tions of u wher e f and h ar e given. If we can use the algor ithm propose d in [10] a nd [22], as w e will see in Remark 18, our deterministic algorithm uses one facto rization of a univ ar iate p olynomial of degree d with algebraic co efficients o f degree at most d , and at most ˜ O ( d n + ω +2 ) if n ≥ 3 or ˜ O ( d 6 ) if n = 2 arithmetic opera tions in K . With the to ols used for the decomp osition alg o rithms, we can compute a L ¨ uroth’s generator . Theorem 4. L et f 1 , . . . , f m ∈ K ( X 1 , . . . , X n ) b e m r ational funct ions of de gr e e at most d . Ther e exists a family of c omput ation tr e es over K p ar ametrize d by z = ( z 1 , . . . , z m ) ∈ K 2 nm , such that: If for al l i = 1 , . . . , m , z i ∈ K 2 n b elongs to an op en Zariski set r elate d t o f 1 , . . . , f i then the tr e e c orr esp onding to z is exe cutable on f 1 , . . . , f m and it r eturns a L¨ ur oth’s gener ator of K ( f 1 , . . . , f m ) . F urt hermor e, we have: (1) An exe cu t able tr e e p erforms 2 m gc d c omputations in K [ X 1 , . . . , X n ] with p olynomials of de gr e e at most d . (2) If K has at le ast (4 d + 2) d elements then we have the estimate: an exe cutable tr e e p erforms ˜ O ( md n ) arithmetic op er ations in K . As b efore, this a lgorithm is softly optimal b e c ause the order of magnitude of the input is md n . A precise description of the open Za riski set is g iven in Remark 29. In the last section w e prov e the following result: Theorem 5. L et f = f 1 /f 2 ∈ K ( X ) . f = u ◦ h , with h = h 1 /h 2 if and only if H ( X , Y ) = h 1 ( X ) h 2 ( Y ) − h 2 ( X ) h 1 ( Y ) divides F ( X , Y ) = f 1 ( X ) f 2 ( Y ) − f 2 ( X ) f 1 ( Y ) . F urt hermor e, if h 1 /h 2 is a r e duc e d n on-c omp osite r ational fun ct ion then H is one of the irr e ducible factors with the smal lest de gr e e re latively to X of F . The fir st pa rt o f this theorem is alr e a dy known, see [30]. Here, we prove that H is irreducible if h 1 /h 2 is non-comp osite. This result implies that we can mo dify the exp onential time decomp osition alg orithm presented in [15] and get a p olynomial time algorithm. Comparison with other algorithms . Ther e already exist several alg orithms for the decomp osition of rationa l functions. In [15], the author s pr ovide t wo algo rithms to decomp ose a multiv ar iate r ational function. These algor ithms run in e xpo nential time in the worst case. In the firs t one we hav e to factorize f 1 ( X ) f 2 ( Y ) − f 1 ( Y ) f 2 ( X ) and to lo ok for factor s of the following kind h 1 ( X ) h 2 ( Y ) − h 1 ( Y ) h 2 ( X ). The a uthors say that in the worst case the n um b er o f candidates to b e tested is ex p o nential in d = deg( f 1 /f 2 ). In the last section we show that a c tually the num b er o f ca ndidates 6 G. CH ` EZE is bounded by d . Th us w e can get a polynomia l time algorithm. In the second a lgorithm, for ea ch pair o f factors ( h 1 , h 2 ) of f 1 and f 2 (i.e. h 1 di- vides f 1 and h 2 divides f 2 ), we ha ve to test if there exists u ∈ K ( T ) such that f 1 /f 2 = u ( h 1 /h 2 ). Th us in the w orst case we also hav e an expone ntial num ber of candidates to be tested. T o the autho r ’s knowledge, the first poly nomial time algo rithm is due to J .-M. Ol- lagnier, see [24] . This algorithm relies on the study o f the k ernel of the following deriv ation: δ ω ( F ) = ω ∧ dF , where F ∈ K [ X ] and ω = f 2 d f 1 − f 1 d f 2 . In [24] the author shows that we can re duce the deco mpo sition of a r ational function to linear alg e br a. The bo ttlenec k o f this a lgorithm is the computation o f the kernel of a matrix. The s iz e of this matrix is O ( d n ) × O ( d n ), then the complexity of this deterministic algor ithm b elongs to O ( d nω ). In [2 4], as in this pap er, the study of the pencil µf 1 − λf 2 plays a crucia l role. Structure of this pap er. In Section 1 , we give a to olb ox where we recall some results ab o ut decompo sition a nd factoriza tion. In Section 2, we descr ibe our al- gorithms to dec o mpo se multiv ar iate ra tio nal functions. In Section 3, w e give an indecomp osability test base d on the study of a Newton’s p olytop e. In Section 4, we give tw o algorithms to compute a L ¨ uroth’s generator . In Section 5 we show tha t the deco mpo s ition algo rithm presented in [15] can be modified to ge t a p olynomial time complexity alg orithm. Notations. All the rational functions are suppo sed to be reduced. Given a p olynomial f , deg ( f ) denotes its total degree. K is an algebraic clo sure of K . F or the sake of simplicity , s ometimes w e wr ite K [ X ] ins tead o f K [ X 1 , . . . , X n ], for n ≥ 2 . Res ( A, B ) denotes the resultant of t wo univ a riate po lynomials A and B . F or any p olynomial P ∈ K [ X ], we write U ( P ) := { a ∈ K n | P ( a ) 6 = 0 } . 1. Prerequisite The fo llowing result implies, as men tioned in the introductio n, that hypo thesis (H) is not restrictive. Lemma 6. If f 1 /f 2 is r e duc e d in K ( X 1 , . . . , X n ) , wher e n ≥ 1 and Λ is a variable, then f 1 + Λ f 2 is squar efr e e. Now we in tro duce our main tools. Prop ositio n 7 . L et f = f 1 /f 2 b e a r ational function in K ( X 1 , . . . , X n ) . f is c omp osite if and only if µf 1 − λf 2 is r e ducible in K [ X ] for al l µ, λ ∈ K su ch that deg ( µf 1 − λf 2 ) = deg( f ) . We also have: f is non-c omp osite if and only if its sp e ctru m σ ( f 1 , f 2 ) is finite, if and only if f 1 − T f 2 is absolutely irr e ducible in K ( T )[ X ] , wher e T is a new variable. F urt hermor e if deg( f ) = d then σ ( f 1 , f 2 ) c ontains at most d 2 − 1 elements. Pr o of. The first par t of this result was known by Poincar´ e see [26], for a mo dern statement and a pro of, see [7, Corollary 2.3]. The b ound | σ ( h 1 , h 2 ) | ≤ d 2 − 1 is proved for a n y field in the biv ar iate ca se in DECOMPOSITION OF M UL TIV ARIA TE RA TIONAL FUNCTIONS 7 [23]. W e deduce the multiv a riate case easily thanks to the Ber tini’s irreducibility theorem, see e.g. [7] or the pro of of Theo rem 13 in [4] for an a pplication of the Bertini’s irreducibility theor em in this con text.  Lemma 8. L et h = h 1 /h 2 b e a r ational function in K ( X ) , u = u 1 /u 2 a r ational function in K ( T ) and set f = u ◦ h with f = f 1 /f 2 ∈ K ( X ) . F or al l λ ∈ K such that deg ( u 1 − λu 2 ) = deg u , we have f 1 − λf 2 = e ( h 1 − t 1 h 2 ) · · · ( h 1 − t k h 2 ) wher e e ∈ K , k = de g u and t i ∈ K a r e t he ro ots of t he u nivariate p olynomial u 1 ( T ) − λu 2 ( T ) . Pr o of. See the pr o of of Lemma 39 in Section 5. Lemma 39 is a g eneralizatio n of Lemma 8. W e state Lemma 8 in o ur to olb ox b ecause the generalization will be only used in Section 5.  R emark 9 . If t i ∈ K then h 1 − t i h 2 ∈ K [ X 1 , . . . , X n ] is an irreducible fa ctor of f 1 − λf 2 . Thus with a rational factor iz a tion we get informa tion a bo ut the decomp osition of f . This re mark will b e us e d during our proba bilistic decomp osition algo rithm in order to a void an absolute factoriza tion. 2. Decomposition algorithms 2.1. Computation of u . Supp ose tha t f = f 1 /f 2 = u ◦ h ∈ K ( X 1 , . . . , X n ), h ∈ K ( X 1 , . . . , X n ), and u ∈ K ( T ). W e set h = h 1 /h 2 . Usually , when h 1 and h 2 are given we get u = u 1 /u 2 by solving a linea r system, see [15, Corollary 2]. Let M ( h 1 , h 2 ) be the matrix c o rresp onding to this linear system in the monomial basis . In our situation the size of M ( h 1 , h 2 ) is O ( d n ) × O ( d ). Thus we can find u with ˜ O ( d n + ω − 1 ) op e r ations in K . W e ca n get u with another approa ch. This appro ach is based on a strategy due to Zipp e l in [41]. Zipp el s how ed in the univ ariate case that we ca n compute u quickly . His strategy is the following: c o mpute the power series H such that h ◦ H ( X ) = X , then compute f ◦ H , and fina lly deduce u with a Pad ´ e approximant. All these steps can b e done with ˜ O ( d ) o r ˜ O ( d 3 / 2 ) arithmetic o p e rations, see [8, Chapter 1 ], and [6]. Thus we deduce tha t in the univ aria te case, u can be computed with ˜ O ( d 3 / 2 ) arithmetic ope r ations. In the multiv a riate ca se with hypothesis (H), we hav e deg( f ) = deg X n ( f ). Th us f (0 , X n ) = u ◦ h (0 , X n ) is no t a consta nt. Then we can apply Zipp el’s s trategy to f (0 , X n ) in order to find u . This method is cor rect b ecause if f and h are given then there is a unique u such that f = u ◦ h , see [15, Corolla ry 2]. Thus we have prov ed the following result: Lemma 10. L et f , h ∈ K ( X 1 , . . . , X n ) b e r ational functions. We supp ose t hat f satisfies hyp othesis (H ) and we set deg( f ) = d . If ther e exists u ∈ K ( T ) such that f = u ◦ h then we c an c ompute u with ˜ O ( d n ) arithmetic op er ations. Pr o of. W e compute f (0 , X n ) with ˜ O ( d n ) arithmetic op era tions. Then we co mpute u as explained abov e with ˜ O ( d 3 / 2 ) arithmetic oper ations.  8 G. CH ` EZE 2.2. A probabili stic algorithm. Decomp Input: f = f 1 /f 2 ∈ K ( X 1 , . . . , X n ), z := ( a , b ) ∈ K 2 n . Output: A decomp osition of f if it exists, with f = u ◦ h , u = u 1 /u 2 , h = h 1 /h 2 non-comp osite and deg u ≥ 2. (1) W e set F a = f 2 ( a ) f 1 ( X ) − f 1 ( a ) f 2 ( X ), F b = f 2 ( b ) f 1 ( X ) − f 1 ( b ) f 2 ( X ). (2) F actoriz e F a and F b . (3) If F a or F b is irreducible then Return “ r is non-comp osite” . (4) Let F a (resp. F b ) be an irreducible factor of F a (resp. F b ) with the smallest degree. (5) Set h = F a / F b . (6) Compute u such that f = u ◦ h as explained in Section 2.1. (7) Return u, h . Exemple 11 . a- W e cons ider f = f 1 /f 2 , with f 1 = X 3 + Y 3 + 1 and f 2 = 3 X Y . W e set a = (0 , 0 ), b = (0 , 1). Then F a = − 3 X Y and F b = 3 X 3 + 3 Y 3 − 6 X Y + 3. F a is reducible but F b is irr e ducible then we co nclude that f is no n-comp osite. b- Now, we apply the algorithm Deco mp to the ra tio nal function f = u ◦ h , where u = ( T 2 + 1) /T and h = h 1 /h 2 with h 1 = X 3 + Y 3 + 1 and h 2 = 3 X Y . W e hav e seen abov e that h is non- c o mpo site. In this situation with a = (0 , 0 ) and b = (0 , 1) we get: F a = − 3 .X .Y . ( X 3 + Y 3 + 1) , and F b = − 12 .X .Y . ( X 3 + Y 3 + 1) . Then the algor ithm cannot give a corr ect output in this situation. Here, we have f 2 ( a ) = f 2 ( b ), w e will see that w e m ust av o id this situa tion. If w e set a = (2 , 1) and b = (1 , − 1) then: F a = 6 0 . ( X 3 + Y 3 − 5 X Y + 1 ) . ( X 3 + Y 3 − 3 5 X Y + 1) , and F b = − 3 . ( X 3 + Y 3 + X Y + 1) . ( X 3 + Y 3 + 3 X Y + 1 ) . Then w e get F a = X 3 + Y 3 − 5 X Y + 1 a nd F b = X 3 + Y 3 + X Y + 1. The a lgorithm Decomp retur ns h = F a / F b . This is a co rrect output since U ◦ F a / F b = h 1 /h 2 , where U =  T / 6 + 5 / 6  /  − T / 2 + 1 / 2  . Prop ositio n 12. If a , b ar e not the ro ots of some non-zer o p olynomials then the algorithm c orr esp onding t o z = ( a , b ) is c orr e ct. Pr o of. First, w e supp ose that f is non-compo site and w e se t Spec t f 1 ,f 2 ( T 1 , T 2 ) = Y ( µ : λ ) ∈ σ ( f 1 ,f 2 ) ( µT 2 − λT 1 ) . W e hav e Sp ect f 1 ,f 2 ( µ, λ ) = 0 if and only if ( µ : λ ) ∈ σ ( f 1 , f 2 ). If Spe c t f 1 ,f 2  f 2 ( a ) , f 1 ( a )  . Spec t f 1 ,f 2  f 2 ( b ) , f 1 ( b )  6 = 0 then F a and F b are absolutely irreducible and deg F a = deg F b = deg f . This giv es: if a and b a void the ro ots of S ( A, B ) := Sp ect f 1 ,f 2  f 2 ( A ) , f 1 ( A )  . Spec t f 1 ,f 2  f 2 ( B ) , f 1 ( B )  , DECOMPOSITION OF M UL TIV ARIA TE RA TIONAL FUNCTIONS 9 where deg S ≤ 2 d ( d 2 − 1) by Propo sition 7, then the algorithm returns: “ r is non- comp osite”. Second, we supp ose f = v ◦ H , with H ∈ K ( X 1 , . . . , X n ) a non-co mpo site rational function. W e set v = v 1 /v 2 , H = H 1 /H 2 such that these tw o rationa l functions are reduced. W e also supp ose that f 2 ( a ) and f 2 ( b ) are nonzero. If deg F a = deg F b = deg f then a and b ar e no t the ro ots of a p olynomia l D of degree d . Thanks to Lemma 8 w e hav e: F a = e ( H 1 − t 1 H 2 ) · · · ( H 1 − t k H 2 ) , F b = e ′ ( H 1 − s 1 H 2 ) · · · ( H 1 − s k H 2 ) , with e, e ′ ∈ K , t i , s j ∈ K . As H 1 ( a ) /H 2 ( a ) (res p. H 1 ( b ) /H 2 ( b )) is a ro ot of f 2 ( a ) v 1 ( T ) − f 1 ( a ) v 2 ( T ) (resp. f 2 ( b ) v 1 ( T ) − f 1 ( b ) v 2 ( T )), we set t 1 = H 1 ( a ) /H 2 ( a ) and s 1 = H 1 ( b ) /H 2 ( b ), and we remark that t 1 , s 1 ∈ K . W e s et Spec t H 1 ,H 2 ( T ) = Y λ ∈ σ ( H 1 ,H 2 ) ∩ K ( T − λ ) . If Sp ect H 1 ,H 2 ( t 1 ) 6 = 0 (r e s p. Sp ect H 1 ,H 2 ( s 1 ) 6 = 0) then H 1 − t 1 H 2 (resp. H 1 − s 1 H 2 ) is absolutely irreducible. If R ( a , b ) = R es T  f 2 ( a ) v 1 ( T ) − f 1 ( a ) v 2 ( T ) , f 2 ( b ) v 1 ( T ) − f 1 ( b ) v 2 ( T )  6 = 0 then t i 6 = s j for all i, j . W e remar k tha t R is a nonzer o poly no mial by Lemma 6 since v 1 and v 2 are copr ime. Thus step 4 gives F a = H 1 − tH 2 , F b = H 1 − s H 2 with t, s ∈ K and t 6 = s . Then h = F a / F b is no n-comp osite, beca us e H 1 /H 2 is non-comp osite.  R emark 13 . Now, with the notations of the pre vious pr o of, we can explain in details the meaning of: “If a , b ar e n ot the r o ots of some non-zer o p olynomials” in Prop ositio n 12 and Theorem 2. This means: If f is non-compo site then there exists a no nzero polyno mial P ( A , B ) := S ( A , B ) of de g ree at most 2 d ( d 2 − 1) such that for any ( a, b ) ∈ U ( P ) the alg orithm co rre- sp onding to z is executable and returns a co rrect output. If f is composite then there exists a nonzero p olynomial D 1 ( A , B ) := f 2 ( A ) .f 2 ( B ) .D ( A ) .D ( B ) of degree at most 4 d s uch that; for any ( a , b ) ∈ U ( D 1 ), there exist nonzero po lynomials D 2 ( A ) := Y λ ∈ σ ( H 1 ,H 2 ) ∩ K  H 2 ( A ) − λH 1 ( A )  of degree at most ( d 2 − 1) .d/ 2 , and R ( A , B ) where deg A R ≤ d 2 / 2 and deg B R ≤ d 2 / 2, such that; fo r any ( a, b ) ∈ U  D 2 ( A ) .D 2 ( B ) .R ( A, B )  , the algor ithm cor resp onding to z = ( a, b ) is ex ecutable and returns a correct output. 10 G. CH ` EZE Prop ositio n 14. Under hyp otheses (C) and ( H ), if a and b ar e not the ro ots of a non-zer o p olynomial then we c an use the algori thm pr op ose d in [22] . Then the algorithm Decomp p erforms one factorization of a univariate p olynomial of de gr e e d over K p lus a numb er of op er ations in K b elonging to ˜ O ( d n ) if n ≥ 3 or to ˜ O ( d 3 ) if n = 2 . Pr o of. As f satisfies (H, i ), w e deduce that if a and b are not the ro ots of a po lyno- mial D of degree d , then the monic part rela tively to X n of F a (resp. F b ) satisfies (L, i ). W e set: D (Λ ) = Res X n  f 1 (0 , X n ) − Λ f 2 (0 , X n ) , ∂ X n f 1 (0 , X n ) − Λ ∂ X n f 2 (0 , X n )  . By hypo thesis (H, ii ), D (Λ) 6 = 0 in K [Λ ]. F urthermore if f 2 ( a ) and f 2 ( b ) ar e nonzer o and D  f 1 ( a ) /f 2 ( a )  6 = 0 (resp. D  f 1 ( b ) /f 2 ( b )  6 = 0 ) then hypothes is (L, ii ) is satis- fied for F a (resp. F b ). Then we can use Lecerf ’s algorithm, see [2 2]. This gives: if a and b av oid the ro ots of D ( A , B ) = D  f 1 ( A ) /f 2 ( A )  . D  f 1 ( B ) /f 2 ( B )  .  f 2 ( A ) .f 2 ( B )  deg D +1 , and deg D ≤ 2  d ( d − 1) d + d  then we can use the algorithm prop osed by G. Lecerf in [22]. The complexity result comes from Lemma 10, and [22, Pro po sition 5], [21, Prop o- sition 2] and [22, Errata].  R emark 15 . The meaning of the condition “if a and b ar e not the r o ots of a non- zer o p olynomial” in Pro po sition 14 is the fo llowing: If we wan t to use Lecerf ’s factorization a lg orithm in order to get the complexit y estimate given in the second part of Theorem 2, then a and b m ust also a void the ro o ts of the po lynomial D ( A ) .D ( B ) . D ( A, B ) , where deg D ≤ d and deg D ≤ 2( d 2 ( d − 1) + d ). It follows that Theor em 2 comes from Propo sition 12 and Pr op osition 14. 2.3. A determi ni stic algorithm. Decom p Det Input: f = f 1 /f 2 ∈ K ( X 1 , . . . , X n ), S = { s 0 , . . . , s B } a s ubset o f K with a t lea st B + 1 = max( d 2 , 3 2 d 2 − 2 d + 1) distinct elemen ts. Output: A decomp osition of f if it exists, with f = u ◦ h , u = u 1 /u 2 , h = h 1 /h 2 non-comp osite and deg u ≥ 2. t:=false, λ := 0. While t=false do (1) If deg( f 1 + s λ f 2 ) = deg( f ) then go to step 2 else λ := λ + 1. (2) Compute the absolute factorization of F λ := f 1 + s λ f 2 . (3) If F λ is absolutely irreducible then Return “ f is non-compo site”. (4) If F λ is absolutely reducible then (a) If t wo distinct abso lute irreducible factors f 1 , f 2 belo ng to K [ X ] then we set h 1 := f 1 and h 2 := f 2 , If ther e exists an absolute ir reducible factor f 1 := F 1 + ǫ F 2 , with ǫ ∈ K \ K and F 1 , F 2 ∈ K [ X ] then w e set h 1 := F 1 , h 2 := F 2 , Else λ := λ + 1 and go to step 1 . DECOMPOSITION OF M UL TIV ARIA TE RA TIONAL FUNCTIONS 11 (b) Co mpute u (if it exists) s uch that f = u ◦ h as explained in Section 2.1. (c) If u exists then t:=true else λ := λ + 1 . Return u, h . Exemple 16 . a- W e c o nsider f = f 1 /f 2 , where f 1 = 3 X Y and f 2 = X 3 + Y 3 + 1 . This gives F 0 = 3 .X .Y , then F 0 is reducible, and this g ives h = X/ Y . W e do not find a rational function u such that f = u ◦ ( X/ Y ) then we consider F 1 = f 1 + f 2 . F 1 is absolutely irreducible, then the algorithm Decomp Det returns f is non-comp osite. b- Now, w e apply the algo rithm Decomp Det to the ra tional function f = u ◦ h , where u = ( T 2 + 1 ) /T a nd h = ( X 3 + Y 3 + 1 ) / (3 X Y ). As we have see n ab ov e h is no n-comp osite. In this situation w e hav e: F 0 = ( X 3 + Y 3 + 1 + 3 .i .X .Y )( X 3 + Y 3 + 1 − 3 .i .X .Y ) , where i 2 = − 1 . Then w e have f 1 = X 3 + Y 3 + 1 + 3 .i.X .Y , F 1 = X 3 + Y 3 + 1, F 2 = 3 X Y . The algorithm returns F 1 / F 2 = h . Prop ositio n 17. The algo rithm is c orr e ct. F urthermor e we go b ack to s t ep 1 at most O ( d 2 ) t imes. Pr o of. First, we supp ose that f is no n-comp osite. By P r op osition 7 there exists s λ 0 ∈ S such that s λ 0 6∈ σ ( f 1 , f 2 ) b eca us e S co n tains at leas t d 2 elements. Thus f 1 + s λ 0 f 2 is absolutely irreducible and step 3 returns f non-comp osite. W e remar k that if f 1 + s λ f 2 is reducible then we cannot find u during step 4b bec ause f is non-comp osite. Then if f is non-comp osite the algorithm is correct. Second, we supp os e that f is compo s ite and f = v ◦ H with H = H 1 /H 2 a reduced and non- comp osite rational function, deg v ≥ 2 and v = v 1 /v 2 is a reduced rational function. f 1 + s λ f 2 = e Q i ( H 1 + t i H 2 ) b y Lemma 8, where ( v 1 + s λ v 2 )( t i ) = 0. There exists s λ 0 ∈ S such that D ( s λ 0 ) 6 = 0, where D (Λ) = Res ( v 1 + Λ v 2 , v ′ 1 + Λ v ′ 2 ) × Y x i ∈ σ ( H 1 ,H 2 ) ∩ K  v 2 ( x i ) − Λ v 1 ( x i )  . Indeed D (Λ) is a nonzer o polyno mia l by Lemma 6 since v 1 and v 2 are co prime. F urthermore , by Prop osition 7, w e ha ve deg D ≤ deg v (deg v − 1) +  (deg H ) 2 − 1  . deg v . As deg v . deg H = d and deg v ≥ 2 , we get deg D ≤ 3 / 2 d 2 − 2 d. As S cont ains at least 3 / 2 d 2 − 2 d + 1 distinct ele men ts, there exists s λ 0 ∈ S such that D ( s λ 0 ) 6 = 0 and then for a ll i , t i 6∈ σ ( H 1 , H 2 ), and t i 6 = t j for all i 6 = j . Then for λ 0 we construct h 1 and h 2 as explained in step 4a. (If t 1 , t 2 ∈ K are distinct then we ha ve tw o absolutely irreducible factors in K [ X ], else if t 1 ∈ K \ K then w e construct h 1 and h 2 with only one absolutely irreducible facto r.) W e ha ve 12 G. CH ` EZE h 1 /h 2 = w ◦ H 1 /H 2 where w ∈ K ( T ) and deg w = 1. W e remark that if f is comp osite then we find a decompo sition f = u ◦ h with h non-comp osite. Indeed, ther e exist ( µ : λ ) and ( µ ′ : λ ′ ) 6 = ( µ : λ ) ∈ P 1 K such that µh 1 + λh 2 and µ ′ h 1 + λ ′ h 2 are a bsolutely ir r educible. (It is obvious if t 1 , t 2 ∈ K . If t 1 ∈ K \ K there exists a conjugate t ′ 1 of t 1 ov e r K suc h that h 1 + t ′ 1 h 2 is abso lutely irreducible.) Then h 1 /h 2 is non comp osite by P r op osition 7. Thus if f is non- comp osite the output is correct.  Theorem 3 is a direct corollary o f Prop osition 17. R emark 18 . In [1 0] the author s s how that we can compute, under the hypothesis (C), the absolute factoriza tio n of a biv ariate squar efree p oly no mial with a t most ˜ O ( d 4 ) arithmetic op er a tions. As we go back to s tep 1 at mo st O ( d 2 ) times we deduce that the algorithm Decomp De t uses at most ˜ O ( d 6 ) arithmetic op erations. When n ≥ 3, a complexity analysis of an absolute fac to rization algo rithm as studied in [1 0] is not done, but we can estimate the cost of our deterministic algorithm. In- deed, w e can reduce absolute factor ization to factor iz ation over a suitable alge braic extension K [ α ] o f deg r ee at most d o ver K , [3 4, 35, 12, 19]. With this stra tegy and with the deterministic factorization a lgorithm prop osed in [2 2] we get an abso lute factorization algo rithm w hich p erforms at most ˜ O ( d n + ω − 1 ) arithmetic op erations in K [ α ]. Thu s the algorithm p erforms ˜ O ( d n + ω ) arithmetic o per ations in K , b ecause [ K [ α ] : K ] ≤ d . As we go back to step 1 at most O ( d 2 ) times w e deduce that, if w e can use Le cerf ’s deterministic facto r ization algorithm, the algorithm Decomp Det uses at most ˜ O ( d n + ω +2 ) arithmetic op erations and one factoriza tion of a univ ar iate po lynomial of degree d with coefficients in K [ α ]. 3. A n indecomposability test using Newton’s pol yto pe In Section 2, if f 1 and f 2 are sparse our alg orithms do no t use this information. In this s e ction we give an indecomp osability test ba s ed o n some prop erties of the Newton’s p olyto pe. The idea is to g eneralize this r emark: if deg f is a prime int eger then f is non-c o mpo site. This is obvious bec ause f = u ◦ h implies deg f = deg u. deg h , a nd deg u ≥ 2. Definition 19. Let f ( X ) ∈ K [ X 1 , . . . , X n ], the supp or t of f ( X ) is the set S f of int eger p oints ( i 1 , . . . , i n ) such that the monomial X i 1 1 · · · X i n n app ears in f with a nonzero co e fficien t. W e denote by N ( f ) the conv ex h ull (in the real space R n ) of S f . This set N ( f ) is called the Newton’s polytop e of f . Definition 20. W e s et N ( f 1 /f 2 ) = N ( f 1 − Λ f 2 ) where Λ is a v ariable, and where f 1 − Λ f 2 is considered as a polynomial with co efficients in K [Λ]. R emark 21 . As Λ is a v aria ble N ( f 1 − Λ f 2 ) is the conv ex hu ll of S f 1 ∪ S f 2 . W e reca ll the classical Bertini-Kr ull’s theorem in our context, see [31, Theorem 37]. Theorem 22. (Bertini-Krul l) L et f 1 /f 2 a r e duc e d r ational fu n ction. Then the fol lowing c onditions ar e e qu ivalent: DECOMPOSITION OF M UL TIV ARIA TE RA TIONAL FUNCTIONS 13 (1) f 1 /f 2 is c omp osite, (2) (a) either ther e exist h 1 , h 2 ∈ K [ X ] with deg X f 1 ( X ) − Λ f 2 ( X ) > max(deg h 1 , deg h 2 ) and a i (Λ) ∈ K [Λ] , such that f 1 ( X ) − Λ f 2 ( X ) = e X i =0 a i (Λ) h 1 ( X ) i h 2 ( X ) e − i ; (b) or the char acteristic p of K is p ositive and f 1 ( X ) − Λ f 2 ( X ) ∈ K [Λ][ X p 1 , . . . , X p n ] . Lemma 2 3. If f 1 /f 2 is a c omp osite r ational funct ion and the char acteristic p of K is such that p = 0 or p > d , then ther e exist e ∈ N , h 1 , h 2 ∈ K [ X ] such that N ( f 1 /f 2 ) = eN ( h 1 /h 2 ) . Pr o of. By Theo rem 2 2 we hav e f 1 ( X ) − Λ f 2 ( X ) = P e i =0 a i (Λ) h 1 ( X ) i h 2 ( X ) e − i . W e denote b y u (Λ , χ ) the poly nomial u (Λ , χ ) = e X i =0 a i (Λ) χ i = a e (Λ) e Y i =1  χ − ϕ i (Λ)  , where ϕ i (Λ) ∈ K (Λ). Thu s f 1 ( X ) − Λ f 2 ( X ) = a e (Λ) e Y i =1  h 1 ( X ) − ϕ i (Λ) h 2 ( X )  . All the factors h 1 ( X ) − ϕ i (Λ) h 2 ( X ) ∈ K (Λ)[ X ] hav e the same suppor t. Indeed, if we supp ose the converse then there exist a coefficient c 1 ∈ K of h 1 and a co efficient c 2 ∈ K of h 2 and tw o indices i and j such that: c 1 − ϕ i (Λ) c 2 = 0 , c 1 − ϕ j (Λ) c 2 6 = 0 . Then c 2 6 = 0 and ϕ i (Λ) = c 1 /c 2 ∈ K . Th us h 1 − ϕ i (Λ) h 2 ∈ K [ X ] is a factor o f f 1 ( X ) − Λ f 2 ( X ). This implies f 1 ( X ) − Λ f 2 ( X ) is reducible in K [Λ][ X ]. This is impo ssible beca us e f 1 and f 2 are coprime. Then, for all i = 1 , . . . , e , we have: N  h 1 − ϕ i (Λ) h 2  = N ( h 1 − Λ h 2 ) = N ( h 1 /h 2 ) . W e reca ll tha t F = F 1 .F 2 implies N ( F ) = N ( F 1 ) + N ( F 2 ), see for example [14, Lemma 5], where the s um is the Mink owski’s sum of conv ex sets. Th us w e hav e: N ( f 1 /f 2 ) = N ( f 1 − Λ f 2 ) = e X i =1 N  h 1 − ϕ i (Λ) h 2  = e X i =1 N ( h 1 /h 2 ) = eN ( h 1 /h 2 ) . This is the desired result.  The pr evious lemma says that if f is comp osite then all the vertices of N ( f ) hav e a co mmo n factor: e . This gives our indecomp osability test designed for sparse po lynomials f 1 and f 2 : Corollary 24 (Indeco mpo sability test) . L et p b e the char acteristic of K , and p = 0 or p > d . L et ( i (1) 1 , . . . , i (1) n ) , . . . , ( i ( k ) 1 , . . . , i ( k ) n ) b e the vertic es of N ( f ) . If gc d( i (1) 1 , . . . , i (1) n , . . . , i ( k ) 1 , . . . , i ( k ) n ) = 1 then f is n on-c omp osite. 14 G. CH ` EZE 4. Comput a tion of a L ¨ uroth’s genera to r In this se c tion we show how to compute a L ¨ uroth’s generator . W e give t wo algorithms. The first one follows the strategy prop osed in [32] for univ ar iate rational functions. The s econd one use s the alg orithm Deco mp and the computation of a greatest common right comp onent o f a univ ar iate rational function. 4.1. Generalization of Sederb erg’s algorithm . In this subsection, w e gener - alize Seder be r g’s a lgorithm. Se der b erg’s algorithm, see [32], is a probabilistic al- gorithm to compute a L ¨ uroth’s gener ator in the univ ariate case. Here, we show that the sa me strateg y w orks in the multiv ar iate case. Our algorithm is also a kind o f pro ba bilistic version of the algorithm pres e n ted in [15]. Indeed, here we compute gcd of p olyno mials of the following kind f 2 ( a ) f 1 ( X ) − f 1 ( a ) f 2 ( X ), where a ∈ K n . In [15], the authors compute gcd of p olynomials o f the following kind f 2 ( Y ) f 1 ( X ) − f 1 ( Y ) f 2 ( X ), wher e Y are new independent v ariables . Sederb erg Generalized Input: f ( X ) = f 1 /f 2 ( X ), g ( X ) = g 1 /g 2 ( X ) ∈ K ( X 1 , . . . , X n ) tw o reduce d rational functions, a , b ∈ K n , n ≥ 2 . Output: h ( X ) ∈ K ( X ) such tha t K ( f , g ) = K ( h ), if h e x ists. (1) F a := f 2 ( a ) f 1 ( X ) − f 1 ( a ) f 2 ( X ), G a := g 2 ( a ) g 1 ( X ) − g 1 ( a ) g 2 ( X ). H a := gcd( F a , G a ). If H a is constant then Return “No L ¨ uroth’s generator”, else go to 2. (2) F b := f 2 ( b ) f 1 ( X ) − f 1 ( b ) f 2 ( X ), G b := g 2 ( b ) g 1 ( X ) − g 1 ( b ) g 2 ( X ). H b := gcd( F b , G b ). If H b is constant then Return “No L ¨ uroth’s generator”, else go to 3. (3) Return h := H a /H b . Exemple 25 . a- W e set f = X , and g = Y , a = (0 , 0), b = (1 , 0). Thu s F a = X , G a = Y and H a = 1. The algorithm Sederb erg Generalized gives K ( f , g ) = K ( X , Y ) has “No L ¨ uroth’s generator”. b- W e co nsider f = U ◦ h and g = V ◦ h where h = ( X 3 + Y 3 + 1 ) / (3 X Y ), U = T 2 / ( T + 1), V = ( T + 2) / ( T 3 + 3 ). h is a no n-comp osite rational function. W e set a = (0 , 0 ), b = (2 , 1). In this situation we ha ve: H a = 3 X Y , and H b = 1 2 . ( X 3 + Y 3 − 5 X Y + 1 ) . The algorithm Sederb erg Generalized returns H a /H b . This is a correct out- put b ecause K ( f , g ) = K ( h ) and h = u ◦ ( H a /H b ) where u is the ra tional function u = (20 T + 1) / (1 2 T ). Now, if we set a = (0 , 0), b = (0 , 1) then we get H a = 3 X Y a nd H b = 12 X Y . In this situation the output H a /H b is not corr ect. W e are in a situatio n where h ( a ) = h ( b ) and we will see that we must a void this situation. Prop ositio n 26. Ther e exists an op en Zariski set U ⊂ K 2 n r elate d to f 1 and f 2 , such that for al l ( a , b ) ∈ U the tr e e c orr esp onding to ( a, b ) is exe cut able on f , g and r etu r n s (if it exists) h such that K ( h ) = K ( f , g ) . In order to prov e this prop osition w e recall some results. DECOMPOSITION OF M UL TIV ARIA TE RA TIONAL FUNCTIONS 15 Definition 2 7. Giv en f 1 , . . . , f m ∈ K ( X ), we s ay that they hav e a common right comp onent (CR C) h , if there a re rationa l functions u i ∈ K ( T ), i = 1 , . . . , m , such that f i = u i ◦ h , and deg u i > 1. h is a grea test common right comp onent (GCR C) o f f 1 , . . . , f m if the u ′ i s hav e not a common right comp onent of deg r ee greater than one. Prop ositio n 2 8 . K ( f 1 , . . . , f m ) = K ( h ) if and only if h is a GCRC of f 1 ,. . . , f m . Pr o of. This pr op osition is prov ed in the univ a r iate case in [1] but the pro o f ca n b e extended to the m ultiv ariate case in a straightforw ard wa y .  Pr o of of Pr op osition 26. Firstly , w e supp ose that there exists a L ¨ uroth’s g e nerator h = h 1 /h 2 , where h 1 /h 2 is r educed. Then, by P r op osition 28, f = u ◦ h and g = v ◦ h where u, v ∈ K ( T ) do not have a common right comp onent of degree grea ter than one. Thus K  u ( T ) , v ( T )  = K ( T ). Then there ex ist Q 1 , Q 2 ∈ K [ U, V ] such that Q 1  u ( T ) , v ( T )  /Q 2  u ( T ) , v ( T )  = T . F urthermore by Lemma 8, F a = f 2 ( a ) f 1 ( X ) − f 1 ( a ) f 2 ( X ) = e Y i  h 1 ( X ) − t i h 2 ( X )  where e ∈ K and t i are the roo ts of f 2 ( a ) u 1 ( T ) − f 1 ( a ) u 2 ( T ) =: u a , and G a = g 2 ( a ) g 1 ( X ) − g 1 ( a ) g 2 ( X ) = e ′ Y i  h 1 ( X ) − s i h 2 ( X )  where e ′ ∈ K and s i are the ro ots of g 2 ( a ) v 1 ( T ) − g 1 ( a ) v 2 ( T ) = : v a . W e get: h ( a ) is a common r o ot of u a and v a . Th us h 1 ( X ) − h ( a ) h 2 ( X ) divides F a and G a . If f 2 ( a ) .g 2 ( a ) .Q 2  u ( h ( a )) , v ( h ( a ))  6 = 0 then h ( a ) is the unique common r o ot of u a and v a . Indee d if there exists another ro ot x such that u a ( x ) = v a ( x ) = 0 , then u  h ( a )  = f 1 ( a ) /f 2 ( a ) = u ( x ) and v  h ( a )  = g 1 ( a ) /g 2 ( a ) = v ( x ). It follows: h ( a ) = Q 1  u ( h ( a )) , v ( h ( a ))  Q 2  u ( h ( a )) , v ( h ( a ))  = Q 1  u ( x ) , v ( x )  Q 2  u ( x ) , v ( x )  = x. Now we r emark that if t 6 = s then gcd( h 1 + th 2 , h 1 + sh 2 ) is constant. W e get then: gcd( F a , G a ) = h 1 ( X ) − h ( a ) h 2 ( X ). In the same w ay: gcd( F b , G b ) = h 1 ( X ) − h ( b ) h 2 ( X ). If h ( a ) 6 = h ( b ), this gives the desired result, becaus e K ( h ) = K ( H ) when H = U ◦ h with U =  T − h ( a )  /  T − h ( b )  . Secondly , we suppose that there does not exist a L ¨ uroth’s generato r. Then we have f = u ◦ h and g = v ◦ H , with h, H ∈ K ( X ) non- c o mpo site a nd algebraic ally indep endent. Thu s F a ( X ) = e. Q i  h 1 ( X ) − t i h 2 ( X )  as b efore, with h 1 ( X ) − t i h 2 ( X ) abso lutely irreducible if t i 6∈ σ ( h 1 , h 2 ). The condition t i 6∈ σ ( h 1 , h 2 ) means R ( a ) = Res T  f 2 ( a ) u 1 ( T ) − f 1 ( a ) u 2 ( T ) , Sp ect h 1 ,h 2 ( T )  6 = 0 , 16 G. CH ` EZE where Spect h 1 ,h 2 ( T ) = Q λ ∈ σ ( h 1 ,h 2 ) ∩ K ( T − λ ). In the same w ay , w e hav e G a = e ′ . Q i ( H 1 ( X ) − s i H 2 ( X )) with H 1 ( X ) − s i H 2 ( X ) abs olutely irreducible if S ( a ) = Res T  g 2 ( a ) v 1 ( T ) − g 1 ( a ) v 2 ( T ) , Sp ect H 1 ,H 2 ( T )  6 = 0 . Thu s F a and G a hav e a non trivial common divisor if and only if there exist t i , s j and α ∈ K \ { 0 } suc h that: ( ⋆ ) α  h 1 ( X ) − t i h 2 ( X )  = H 1 ( X ) − s j H 2 ( X ) . In the same w ay , F b and G b hav e a non trivial common div isor if and o nly if there exists t ′ i , s ′ j and α ′ ∈ K \ { 0 } such that: ( ⋆⋆ ) α ′  h 1 ( X ) − t ′ i h 2 ( X )  = H 1 ( X ) − s ′ j H 2 ( X ) . ( ⋆ ) and ( ⋆⋆ ) giv e:  α − αt i α ′ − α ′ t ′ i   h 1 h 2  =  1 − s j 1 − s ′ j   H 1 H 2  . If D ( a , b ) = R es T  g 2 ( a ) v 1 ( T ) − g 1 ( a ) v 2 ( T ) , g 2 ( b ) v 1 ( T ) − g 1 ( b ) v 2 ( T )  6 = 0 then s j 6 = s ′ j and the previo us system giv es H = u ◦ h , with deg u = 1 . Thus h and H are alg ebraically dep endent and this is absurd. Th us F a and G a (resp. F b and G b ) hav e no common divisor. Hence, if no L ¨ uroth’s generator exists and f 2 ( a ) .g 2 ( b ) .R ( a ) .S ( a ) .R ( b ) .S ( b ) .D ( a, b ) is not equal to zero, then gcd( F a , G a ) is constant and gcd( F b , G b ) is constant. Thus the algorithm returns “No L¨ uroth’s generator” .  R emark 2 9 . With the notations o f the pr evious pro o f, we remark that a and b must av o id the ro o ts of: f 2 ( X ), g 2 ( X ), h 2 ( X ), Q 2  f ( X )  , g ( X )  , R ( X ), S ( X ), and ( a , b ) m ust av o id the ro o ts of h 1 ( A ) h 2 ( B ) − h 1 ( B ) h 2 ( A ) and D ( A, B ). W e ca n easily b ound the deg ree of ea ch p olyno mial: deg f i ≤ d , deg g i ≤ d , deg h i ≤ d/ 2, deg Q 2 ≤ d ( d − 1) see [5, Prop ositio n 2.1 ], deg R ≤ d ( d 2 − 1), deg S ≤ d ( d 2 − 1), and deg D ≤ d 3 . Then if K is “ big enough” the open Zaris ki s et U is not the empt y s e t. R emark 3 0 . In the alg orithm Sederb erg Generali zed we cannot co ns ider tw o random linear combinations of f 1 , f 2 and g 1 , g 2 . Indeed, with ra ndo m linear com binations and with the notations of the pr evious pro of, u a and v a do no t hav e a unique common ro ot in K . Th us with random linear co mb inations the strategy used in Prop ositio n 26 is not v alid. Prop ositio n 31 . If K is a field with at le ast (4 d + 2) d elements then t he algorithm Sederb erg Generalized u s es ˜ O ( d n ) arithmetic op er ations. Pr o of. The computations o f f i ( a ), g i ( a ), f i ( b ), g i ( b ) needs ˜ O ( d n ) a rithmetic op er- ations. The complexity of a n n -v aria te gcd computation needs ˜ O ( d n ) arithmetic op erations. Indeed, as K is a field with at le ast (4 d + 2) d e le men ts with Lemma 6.44 in [39] we ca n ge neralize to n v ar ia bles the algo r ithm 6.36 presented in [39] and obtain a result lik e Corollary 11.9 in [3 9]. This gives the desired result.  DECOMPOSITION OF M UL TIV ARIA TE RA TIONAL FUNCTIONS 17 R emark 32 . When it is p ossible, a p o ly nomial gener ator is desirable. The algo r ithm Sederb erg Generalized alwa ys returns a rational g enerator. W e ca n test if we hav e a po lynomial generato r in the following wa y: W e tes t if there e x ist α, β ∈ K such that αH a + β = H b . If such consta n ts exist then H a (or H b ) is a polynomial genera tor. This improv ement is correct b eca use w e hav e se e n during the proo f of Prop osition 26 that H a = h 1 − h ( a ) h 2 and H b = h 1 − h ( b ) h 2 . Th us if a polyno mial gener ator h 1 exists w e ha ve H a = h 1 − h 1 ( a ) and H b = h 1 − h 1 ( b ). As gcd a re known up to a m ultiplicative constant there exist α, β ∈ K such that αH a + β = H b . Conv ersely , if we hav e αH a + β = H b then H a /H b = u ◦ H a with u = T /  αT + β  , thus K ( H a /H b ) = K ( H a ). The computation of α and β needs O ( d n ) arithmetic op era tions. I ndee d, we solve a linear system with O ( d n ) equations and t wo unknowns. Thus we can find a p oly - nomial generator with the algo rithm Sederberg Generali zed with ˜ O ( d n ) arithmetic op erations. 4.2. Another s trategy based on decomp ositi on. Now, we give another alg o- rithm to compute a L¨ uroth’s generator . Here we use the relatio n be tw een dec o m- po sition and computation of a L ¨ uroth’s generator . L¨ uroth with Decomp Input: f ( X ) = f 1 /f 2 ( X ), g ( X ) = g 1 /g 2 ( X ) ∈ K ( X 1 , . . . , X n ) tw o reduce d rational functions, z := ( a, b ) ∈ K 2 n . Output: h ( X ) ∈ K ( X ) such that K ( f , g ) = K ( h ), if h exists. (1) Decomp ose f with the algor ithm Decomp , then f = u ◦ h . (2) Compute v (if it exists) such that g = v ◦ h . (3) If v do not exist then Return “ No L¨ uroth’s generator ”, else g o to 4. (4) Compute w the GCRC of u and v with Seder b er g’s algorithm. (5) Return w ◦ h . Prop ositio n 33. The algo rithm L ¨ uroth’s with Decomp is c orr e ct for z satisfying the hyp othesis of The or em 2. Pr o of. This algorithm computes a GCR C of f and g , thus b y Prop os ition 2 8, this gives the desired result.  Prop ositio n 34. Under hyp otheses (C) and (H), t he algorithm L ¨ uroth’s with De- comp p erforms one factorizatio n of a univariate p olynomial of de gr e e d over K plus a nu mb er of op er ations in K b elonging to ˜ O ( d n ) if n ≥ 3 or to ˜ O ( d 3 ) if n = 2 . Pr o of. The firs t step of the algorithm perfor ms o ne factoriza tion of a univ a riate po lynomial of degree d ov er K plus a num b er o f op eratio ns in K b elonging to ˜ O ( d n ) if n ≥ 3 or to ˜ O ( d 3 ) if n = 2 by Prop o sition 14. With the str ategy prese nted in Section 2.1, the second s tep can b e done with ˜ O ( d n ) arithmetic ope r ations. The last step can b e done in an efficient probabilis tic w ay , see [32]. The algorithm presented in [3 2] computes only t wo gcd’s of univ ar iate po lynomials of degree low e r than d . Then the tota l cos t of the algo r ithm b elongs to ˜ O ( d n ) if n ≥ 3 or to ˜ O ( d 3 ) if n = 2 .  R emark 35 . During the a lgorithm L ¨ uroth with Decomp we hav e to avoid the ro ots of nonzero p olyno mials considered in Remark 1 3 and Remark 15 b eca use we use 18 G. CH ` EZE the algorithm Decomp . F urthermore during the algor ithm L ¨ uroth with Dec omp , w e use Sederb erg ’s alg o rithm, this algorithm is also probabilistic and has in input t wo parameters x 1 , x 2 ∈ K . If x 1 and x 2 are not the roo ts of a nonzero p olyno mials then the output is correct, see [32]. Thu s the no nzero po lynomials are just the ones used for the algorithm Decomp and for Sederb erg’s algorithm. 4.3. Computation of a L ¨ uroth’s generator. L¨ uroth’s generato r Input: f 1 ( X ) , . . . , f m ( X ) ∈ K ( X ), m reduced rational functions, z := z 2 , . . . , z m ∈ K 2 n , n ≥ 2. Output: h ( X ) ∈ K ( X ) such that K ( f 1 , . . . , f m ) = K ( h ), if h exists. (1) Compute a L¨ uroth’s generato r of K ( f 1 , f 2 ) with Sederb erg Generalize d ap- plied to f 1 , f 2 , with z 2 . (2) If a L ¨ uroth’s gener ator h is found then go to step 3 else Return “No L ¨ uroth’s generator ”. (3) F or i = 3 , . . . , m , (a) Compute a L ¨ uroth’s generato r of K ( h, f i ) with Sederb erg Generalize d applied to h , f i , with z i . (b) If a L ¨ uroth’s g enerator H is found then h := H else Return “ No L ¨ uroth’s genera tor”. (4) Return h. Prop ositio n 36. The algo rithm L ¨ uroth’s generator is c orr e ct for z satisfying the hyp othesis of The or em 4. Pr o of. W e just hav e to remark that K ( f 1 , . . . , f i − 1 , f i ) = K ( f 1 , . . . , f i − 1 )( f i ).  Prop ositio n 37. If K has at le ast (4 d + 2 ) d elements, then the algorithm L ¨ uroth’s generator c an b e p erforme d with ˜ O ( md n ) arithmetic op er ations in K . Pr o of. W e use m times the algo rithm Sed erberg Generalized . Thus, thanks to Pro p o - sition 31 w e get the desired c omplexity .  R emark 38 . During the algor ithm L¨ uroth’s generator w e can us e the alg orithm L¨ uro th wi th Decomp instead of Sederb erg Generali zed . In the biv a r iate case, the complexity b ecomes then ˜ O ( d 3 ). In this case the alg orithm is not softly optimal, but the algorithm can a lso return u suc h that f = u ◦ h . W e conclude that Prop osition 36 and Prop osition 37 pro ve Theorem 4. 5. Study of the Gutierez-Rubio-Sevilla ’s algorithm In this section we s tudy the complexity of the decomp osition algorithm given in [15]. More precisely , we explain ho w to mo dify it in order to get a polyno mial time algorithm instead of an exponential time algorithm. 5.1. Some prelimi nary res ults. The following le mma is a g eneralizatio n o f Lemma 8. Lemma 39. L et h = h 1 /h 2 b e a r ational fun ction in K ( X ) , u = u 1 /u 2 a r ational function in K ( T ) and set f = u ◦ h with f = f 1 /f 2 ∈ K ( X ) . L et λ, µ ∈ L , wher e L is a field and K ⊂ L . We have: µf 1 − λf 2 = ( µu 1 − λu 2 )( h ) .h deg u 2 . DECOMPOSITION OF M UL TIV ARIA TE RA TIONAL FUNCTIONS 19 Pr o of. W e hav e µf 1 − λf 2 f 2 = µ u 1 ( h ) u 2 ( h ) − λ u 2 ( h ) u 2 ( h ) = µu 1 ( h ) − λu 2 ( h ) u 2 ( h ) . Thu s: ( ⋆ ) ( µf 1 − λf 2 ) .u 2 ( h ) = ( µu 1 − λu 2 )( h ) .f 2 . F urthermore ( ⋆⋆ ) f 1 f 2 = u 1 ( h ) u 2 ( h ) =  P d 1 i =0 a i h i 1 h d 1 − i 2  .h d 2 2  P d 2 i =0 b i h i 1 h d 2 − i 2  .h d 1 2 , where u 1 ( T ) = P d 1 i =0 a i T i , u 2 ( T ) = P d 2 i =0 b i T i . Then f 2 =  P d 2 i =0 b i h i 1 h d 2 − i 2  .h max( d 1 − d 2 , 0) 2 bec ause f is reduced a nd the deg ree of the right term of ( ⋆⋆ ) is lo wer or equal to deg( f ). It follows f 2 = u 2 ( h ) .h max( d 1 − d 2 , 0)+ d 2 2 = u 2 ( h ) .h deg u 2 , then thanks to ( ⋆ ) w e deduce the desired result.  Prop ositio n 40. L et f ∈ K ( X ) b e a ra tional function such that f = u ◦ h and f = u ◦ ϕ , wher e u is a ra tional function in K ( T ) , h a non- c omp osite r ational function and ϕ a r ational function. Then ϕ is non- c omp osite and t her e exists w ∈ K ( T ) su ch that h = w ◦ ϕ and deg w = 1 . R emark 41 . w is not necessarily the identit y . F or example if u = x 2 + 1 /x 2 and w = 1 /x then u ◦ w = u . Thus we ca n get f = ( u ◦ w ) ◦ ϕ = u ◦ ϕ and f = u ◦ ( w ◦ ϕ ) = u ◦ h . See [16] for more statemen ts on the particular situation u ◦ w = u . Pr o of. W e set u = u 1 /u 2 and ϕ = ϕ 1 /ϕ 2 . Let λ, µ ∈ K such that deg( µu 1 − λu 2 ) = deg u , b y Lemma 39 we have µf 1 − λf 2 = e deg u Y i =1 ( h 1 − x i h 2 ) , where e ∈ K and x i ∈ K are the ro ots of µu 1 − λu 2 . W e can suppo se that h 1 − x i h 2 are absolutely irreducible and x i 6 = x j if i 6 = j . Indeed, the “bad” v alues o f ( µ : λ ) ar e ( u 2 ( x ) : u 1 ( x )) wher e x ∈ σ ( h 1 , h 2 ) and are the r o ots of R ( µ, λ ) = Res ( µu 1 − λu 2 , µu ′ 1 − λu ′ 2 ). As σ ( h 1 , h 2 ) is finite and K infinite, w e deduce that “go o d” v a lues of ( µ : λ ) exist. W e can also s uppo se that deg ϕ 1 − x i ϕ 2 = deg ϕ , b ecaus e we just have to av oid a finite n umber of x i . Then Lemma 39 a lso implies µf 1 − λf 2 = e deg u Y i =1 ( ϕ 1 − x i ϕ 2 ) . W e hav e ϕ 1 − x i ϕ 2 is abso lutely irreducible, else µf 1 − λf 2 has more than deg u absolute ir reducible factors: this is a contradiction with h 1 − x i h 2 being absolutely irreducible. Then ϕ is non-compo site by Prop ositio n 7. F urthermore , there exis t i k , j k , with k = 1 , . . . , de g u s uc h that h 1 − x i k h 2 equal ϕ 1 − x j k ϕ 2 up to a multiplicativ e co nstant. As in the pr o of of Prop osition 26 it follows ϕ = w ◦ h with w ∈ K ( T ) and deg w = 1. As h and ϕ b elongs to K ( X ) we hav e w ∈ K ( T ). (Indeed we just have to solve a linear system in K to get w .)  20 G. CH ` EZE 5.2. Study of the absol ute irreducible factors of near-separated p olynomi- als. The decomp osition algor ithm given in [1 5] is based on the follo wing theorem; see [30]. In this subsection w e improv e this result. Theorem 42. L et f = f 1 /f 2 ∈ K ( X ) . f = u ◦ h , with h = h 1 /h 2 if and only if H ( X , Y ) = h 1 ( X ) h 2 ( Y ) − h 2 ( X ) h 1 ( Y ) divides F ( X , Y ) = f 1 ( X ) f 2 ( Y ) − f 2 ( X ) f 1 ( Y ) . In the following we use a result due to Schinzel. Definition 43. A r ational function is reducible ov er K if the numerator in its reduced form is reducible o ver K . Lemma 44. L et Ψ( T , Y ) and f ( X ) b e non-c onstant r ational functions over K , the former of non-ne gative de gr e e with re sp e ct to T and to at le ast one Y i . If the function ψ  f ( X ) , Y  is r e ducible over K then f = u ◦ h , u ∈ K ( T ) , h ∈ K ( X ) and ψ  u ( T ) , Y  is r e ducible over K . Pr o of. See [29, Lemma 1 ].  Prop ositio n 45. L et f = f 1 /f 2 ∈ K ( X ) , ˆ f = ˆ f 1 / ˆ f 2 ∈ K ( Y ) b e two non-c onstant r ational functions. If f and ˆ f ar e non- c omp osite t hen F ( X , Y ) = f 1 ( X ) ˆ f 2 ( Y ) − f 2 ( X ) ˆ f 1 ( Y ) is irr e- ducible in K [ X , Y ] . Pr o of. W e set ψ ( T , Y ) = ˆ f ( Y ) − T . Then ψ  f ( X ) , Y  = ˆ f 1 ( Y ) f 2 ( X ) − f 1 ( X ) ˆ f 2 ( Y ) f 2 ( X ) ˆ f 2 ( Y ) . If we supp ose F ( X , Y ) reducible then f = u ◦ h and ψ  u ( T ) , Y ) is reducible by Lemma 44. As f is non-comp osite deg u = 1 thu s we can set u ( T ) = ( aT + b ) / ( αT + β ). Then ψ  u ( T ) , Y ) is reducible means ˆ f 1 ( Y )( αT + β ) − ˆ f 2 ( Y )( aT + b ) is reducible o ver K . B y Prop ositio n 7 this is a bsurd b ecause ˆ f is non-co mpo site. Hence F ( X , Y ) is irreducible.  Now we can impro ve Theorem 42. Theorem 46. L et f = f 1 /f 2 ∈ K ( X ) a non-c onstant r ational funct ion. If f = u ◦ h , wher e u = u 1 /u 2 ∈ K ( T ) and h = h 1 /h 2 ∈ K ( X ) ar e r ational functions, with deg u ≥ 2 and h non-c omp osite, then the irr e ducible factors with t he smal lest de gr e e r elatively to X of F ( X , Y ) = f 1 ( X ) f 2 ( Y ) − f 2 ( X ) f 1 ( Y ) ar e of the kind H ( X , Y ) = h 1 ( X ) ϕ i, 2 ( Y ) − h 2 ( X ) ϕ i, 1 ( Y ) , wher e ϕ i = ϕ i, 1 /ϕ i, 2 ar e non-c omp osite r ational functions such that h = w ◦ ϕ i with deg w = 1 . Theorem 5 is a direct consequence of Theo rem 46. DECOMPOSITION OF M UL TIV ARIA TE RA TIONAL FUNCTIONS 21 Pr o of. By Lemma 39, w e hav e ( ⋆ ) F ( X , Y ) = U f 1 ,f 2  h ( X )  .h 2 ( X ) deg u , where U f 1 ,f 2 ( T ) = f 2 ( Y ) u 1 ( T ) − f 1 ( Y ) u 2 ( T ) . As f = u ◦ h , h ( Y ) is a roo t of U f 1 ,f 2 . Then U f 1 ,f 2 ( T ) =  h 2 ( Y ) T − h 1 ( Y )  A ( Y , T ) , where A ( Y , T ) ∈ K [ Y , T ]. Thus ( ⋆ ) implies h 1 ( X ) h 2 ( Y ) − h 2 ( X ) h 1 ( Y ) divides F ( X , Y ). Now, we s uppo s e that ϕ ( Y ) ∈ K ( Y ) is another ro o t of U f 1 ,f 2 ( T ). Then u  ϕ ( Y )  = f ( Y ) = u  h ( Y )  . Thu s, by Lemma 40, we hav e ϕ is non-comp os ite and h = w ◦ ϕ with deg w = 1. As befo re, we c an write U f 1 ,f 2 =  ϕ 2 ( Y ) T − ϕ 1 ( Y )  .B ( Y , T ), where B ( Y , T ) ∈ K [ Y , T ]. Thu s ϕ 2 ( Y ) h 1 ( X ) − ϕ 1 ( Y ) h 2 ( X ) divides F ( X , Y ) by ( ⋆ ). Now, we wr ite ( ⋆⋆ ) U f 1 ,f 2 ( T ) = Y i ∈ I  ϕ i, 2 ( Y ) T − ϕ i, 1 ( Y )  . Y j ∈ J C e j j ( Y , T ) , where ϕ i = ϕ i, 1 /ϕ i, 2 ( Y ) is a r e duce d non- comp osite rational function as explained ab ov e a nd C j ( Y , T ) ∈ K [ Y , T ] is irreducible with deg T C j ≥ 2 . W e ev aluate T to h in ( ⋆⋆ ) and m ultiply the result b y h deg u 2 : U f 1 ,f 2  h ( X )  .h 2 ( X ) deg u = Y i ∈ I  ϕ i, 2 ( Y ) h 1 ( X ) − ϕ i, 1 ( Y ) h 2 ( X )  ×  Y j ∈ J C e j j  Y , h ( X )   .h 2 ( X ) P j ∈ J e j deg T C j . The fa ctors ϕ i, 2 ( Y ) h 1 ( X ) − ϕ i, 1 ( Y ) h 2 ( X ) are irreducible by P rop osition 4 5. F ur- thermore, by Lemma 44 as h is non-c o mpo site and C j ( Y , T ) is ir reducible, we have C j  Y , h ( X )  .h 2 ( X ) deg T C j is irreducible in K [ X , Y ]. W e also ha ve deg X C j  Y , h ( X )  h 2 ( X ) deg T C j = deg T C j . deg h ≥ 2 deg h > deg X ϕ i, 2 ( Y ) h 1 ( X ) − ϕ i, 1 ( Y ) h 2 ( X ) . Then H ( X , Y ) = ϕ i, 2 ( Y ) h 1 ( X ) − ϕ i, 1 ( Y ) h 2 ( X ) ar e the factors with the s ma llest degree relatively to X .  5.3. Impro v ement of the GRS algorithm. Now we des crib e the deco mp os ition algorithm presented in [15]. GRS decomp osition algo rithm Input: f ( X ) = f 1 /f 2 ( X ), n ≥ 2. Output: u ∈ K ( T ), h ( X ) ∈ K ( X ) such that f = u ◦ h , or “ f is non-compos ite”. (1) F actor F ( X , Y ). Let D = { H 1 , . . . , H m } be the set of factors of F (up to pro duct by consta n ts). W e set i = 1. 22 G. CH ` EZE (2) If H i can be written H i ( X , Y ) = h 1 ( X ) h 2 ( Y ) − h 1 ( Y ) h 2 ( X ) then h 1 /h 2 is a right compone nt for f . Then co mpute u b y solving a linea r sys tem a nd Return u , h . (3) If i < m then i := i + 1 and go to s tep 2, else Return “ f is non-compo site”. This alg o rithm has an exp onential time complexity . Indeed, the set D contains at most 2 d po lynomials, where d is the degree of f . How ever, w e can improve this algo rithm. Thank s to Prop os itio n 4 5, w e remar k that if f is non-comp osite then F is irreducible. F ur thermore, if f = u ◦ h with h non-comp osite, then H ( X , Y ) = h 1 ( X ) h 2 ( Y ) − h 1 ( X ) h 2 ( Y ) is a n irreducible factor of F ( X , Y ), by Theorem 46. Th us we have to study at most deg F ir reducible factors. T hus we ca n subs titute the s e t D b y the set o f irr e ducible factor s. (W e ca n also substitute the set D by the set of irr e ducible factors with the smal lest de gr e e r elatively to X ). As Step 1 and Step 2 can b e done in a p olynomia l time, it follows: Prop ositio n 47. If in t he GRS deco mpo sition a lgorithm we set: “ D is the set of irr e ducible factors of F ”, t hen this m o difie d algorithm has a p olynomial t ime c omplexity. R emark 48 . The bottleneck o f this modified algor ithm is the fac torization of F . If we apply the deterministic a lgorithm pr o po sed in [2 2] then the mo dified GRS decomp osition algor ithm uses ˜ O ( d 2 n + ω − 1 ) ar ithmetic op erations, where d is the degree of f and n the n umber of v ar iables. Exemple 49 . Now, we illustrate the GRS deco mp osition algorithm with f = u ◦ h , where u = ( T 2 + 1 ) /T , h = h 1 /h 2 , a nd h 1 = X 3 1 + X 3 2 + 1 , h 2 = 3 X 1 X 2 . h is a non-comp osite rational function. In this situation, w e ha ve the following factorization of F ( X 1 , X 2 , Y 1 , Y 2 ): F ( X 1 , X 2 , Y 1 , Y 2 ) = 3 .H 1 ( X 1 , X 2 , Y 1 , Y 2 ) .H 2 ( X 1 , X 2 , Y 1 , Y 2 ) , where H 1 ( X 1 , X 2 , Y 1 , Y 2 ) = X 3 1 Y 1 Y 2 + X 3 2 Y 1 Y 2 + Y 1 Y 2 − Y 3 1 X 1 X 2 − Y 3 2 X 1 X 2 − X 1 X 2 = h 1 ( X 1 , X 2 ) h 2 ( Y 1 , Y 2 ) − h 1 ( Y 1 , Y 2 ) h 2 ( X 1 , X 2 ) , H 2 ( X 1 , X 2 , Y 1 , Y 2 ) = 1 + X 3 1 + X 3 2 + Y 3 1 + Y 3 2 + X 3 1 Y 3 1 + X 3 1 Y 3 2 + X 3 2 Y 3 1 + X 3 2 Y 3 2 − 9 X 1 X 2 Y 1 Y 2 = h 1 ( X 1 , X 2 ) h 1 ( Y 1 , Y 2 ) − h 2 ( Y 1 , Y 2 ) h 2 ( X 1 , X 2 ) . Then we can recov er the decomp ositio n f = u ◦ h with the GRS deco mp osition algorithm . References [1] Cesar Alonso, Jaime Gutierrez, and T omas Recio. A rational function decomp osition algo- rithm by near-separated p olynomials. J. Symb olic Comput. , 19(6):527–544, 1995. [2] Shreeram S. Abh yank ar, William J. H einzer, and A vinash Satha y e. T ranslates of polynomial s . 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