Classification of 3-dimensional integrable scalar discrete equations

Classification of 3-dimensional in tegrable scalar discrete equations ∗ S.P . Tsarev † and T. W olf Octob er 22, 20 1 8 Departmen t of Mathematics T ec hnisc he Unive rsit¨ at Berlin Berlin, German y and Departmen t of Mathematics, Bro c k Univ ersit y 500 G le nridge Av en ue, St.Catharines, On tario, Canada L2S 3A1 e-mails: tsarev@math .tu-berlin.de sptsarev@ma il.ru twolf@brock u.ca Abstract W e classify all in tegrable 3-dimensional s c alar discrete quasilinear equations Q 3 = 0 on an elemen tary cubic cell of the lattice Z 3 . An equation Q 3 = 0 is called in tegrable if it ma y b e consistent ly imp osed on all 3-dimensional elementa ry faces of the lattice Z 4 . Under the natur al requ i rement of inv ariance of the equation under the action of the complete group of symm etries of th e cub e w e prov e that the only nontrivial (non-linearizable) in tegrable equation from this class is the w ell-kno wn dBKP-sys t em. MSC : 37K10, 52C99 Keyw ords : int egrable systems, discrete equations, large p olynomial systems, computer algebra, Re duce , Form , Crack . ∗ Suppo rted b y the DF G Resear ch Unit 565 “Polyhedral Surfac e s ” (TU-Berlin) † On leav e fro m: Krasnoyarsk State Pedagogical Universit y , Russia. SPT ackno wledges partia l financial supp ort from the grant of Siberia n F ederal Universit y (NM-pr o ject N o 45.200 7) and the RFBR grant 0 6-01- 0081 4. 1 1 In t ro ductio n Although definitiv ely shaped a decade ago, discrete differen t ia l geometry (see e.g. [3 , 4]) has already pro vided muc h insigh t in to structures tha t are fundamental b oth to classical differen t ia l geometry a nd to the theory of integrable PDEs. In a ddition to suc h purely mathematical fields, results in discrete differen tial g eometry hav e a great p otential in computer gr aphics and ar chitectural design: it turns out t ha t discrete surfaces pa ram- eterized by discrete conjugate lines and discrete curv ature lines — the basic structures in discrete differen tial geometry — ha v e sup erior approximation prop erties and o t her useful features (see [9 ]). 0 f 00 f 01 f 10 f 1 , − 1 f 0 , − 1 f − 1 , − 1 1 1 x 1 f 11 x 2 Figure 1: Z 2 lattice In this pap er w e consider the cubic lat t ice Z n with v ertices at inte ger p oin ts in the n -dimensional space R n = { ( x 1 , . . . , x n ) | x s ∈ R }} . With eac h v ertex (with in teger co ordinates ( i 1 , . . . , i n ), i s ∈ Z ) w e asso ciate a scalar field v ariable f i 1 ...i n ∈ C . In what follows w e need to consider the elemen tary cubic cell K n = { ( i 1 , . . . , i n ) | i s ∈ { 0 , 1 }} of the lattice Z n . The field v ariables f i 1 ...i n are asso ciated to its 2 n v ertices. W e will use the short notation f for the set ( f 00 ... 0 , . . . , f 11 ... 1 ) o f all these 2 n v ariables. An n -dimensional discrete system of the type considered here is given by an equation of the form Q n ( f ) = 0 , (1) on the field v ariables on the elemen tary cubic cell K n . F or the other elemen tary cubic cells of Z n the equation is the same, after shifting t he indices of f suitably (see Figure 1). In the last t w o decades the study of sp ecial classes of (1) whic h are “integrable” (in one sense or ano t her) has b ecome v ery p opular. W e g ive b elow only a brief a ccount o f the curren t state of this field of researc h, for a more detailed accoun t cf. [1]–[4] and the references given therein. In fact, discrete in tegrable systems underlie many classical in tegrable nonlinear PDEs, lik e the Kric heve r-Novik ov equation and o t her examples, 2 the latter app ear as a con tin uous limit along some of the discrete directions. Other w ell kno wn classes of in tegrable geometric ob jects (with n = 3), like minimal surfaces, conjugate nets, constan t curv ature surfaces, Moutard nets, isothermic surfaces, orthog- onal curvilinear co ordinates etc., are also o btained as some smo oth limits along any t w o of the three directions of the respectiv e discrete system. The remaining third dis- crete direction aut o matically prov ides us with a tra nsformation known in the classical con tin uous geometric con text a s Jonas/Ribaucour/B¨ ack lund transformation b etw een surfaces of the giv en class (see [1 ]–[4] f or more details). On the other hand, starting from the classical theorems on non-linear sup erp osition principles a nd p erm utability of the af oremen tioned transformations b etw een smo oth surfaces of o ne o f these types w e obtain precisely the underlying discrete system. One of the cornerstones of the discrete differen tial geometry (the idea to lo ok for cubic nonlinear sup erp osition f o r- m ulas of B¨ ac klund transformations of nonlinear in tegrable PDEs) was laid dow n in [5]. The dualit y b et w een the smo o th ob jects in an y of the geometric classes of integrable smo oth surfaces men tioned ab ov e and their B¨ ac klund-t yp e tra nsfor ma t io ns is therefore put in to a symmetric form of a single discrete n -dimensional system and is enco ded as the notio n of ( n + 1) -dimensiona l c on sistency [3]: A n n -dimension al discr ete e quation (1) is c al le d c onsistent, if it may b e imp os e d in a c onsis tent way on al l n -dimensional fac es of a ( n + 1) -dime nsional cub e . This can b e also understo o d as the p ossibilit y to take Z n +1 and prescrib e the n - dimensional equation (1) to hold on ev ery n -dimensional face of every elemen tary ( n + 1)-dimensional cub e (of size 1, with edges parallel to the co ordinate a xes) without side relations to app ear. F or this reason (1) is often called a “fa ce form ula”. A precise definition of consistency , suitable fo r the class o f discrete equations treated in this pap er, will b e fo rm ulated in the next section. This pap er is dev oted to application o f computer algebra systems Reduce and F orm ([10]), in particular the Reduce pac k age Crack ([11, 12]), to the classification of 3- dimensional integrable discrete systems. The pap er is organized as f ollo ws. In Section 2 we give a brief description of the kno wn results on 2 -dimensional in tegrable scalar discrete equations of type ( 1 ) and the precise definition of ( n + 1)-dimensional consistency conditio n for such discrete n -dimensional systems . Section 3 is dev oted to the classification of symmetry types of quasilinear equations (1) for dimensions n = 2 , 3 , 4. In Section 4 w e describ e the results of our computations (Theorem 2): the only non trivial (non-linearizable) integrable scalar quasilinear 3 d -face equation in v ariant w.r.t. the complete gr oup of symmetries of the cub e is given by the form ula (5) b elow . App endices A–E describ e the tec hnical details of t he computations. 3 2 The setu p The simplest but v ery imp ortant class of 2- dimensional in tegrable fa ce form ulas w as in v estigated in detail in [1, 2]. They hav e the form Q ( f 00 , f 10 , f 01 , f 11 ) = 0 , (2) where f ij are scalar fields attached to the v ertices of a square (see Fig . 2) with tw o main requiremen ts: 1) Quasilinearit y . (2) is affine linear w.r.t. eve ry f ij , i.e. Q has degree 1 in an y of its four v aria bles: Q = c 1 ( f 10 , f 01 , f 11 ) f 00 + c 2 ( f 10 , f 01 , f 11 ) = c 3 ( f 00 , f 01 , f 11 ) f 10 + c 4 ( f 00 , f 01 , f 11 ) = . . . = q 1111 f 00 f 10 f 01 f 11 + q 1110 f 00 f 10 f 01 + q 1101 f 00 f 10 f 11 + . . . + q 0000 . 2) Symmetry . Equation (2) should b e inv a rian t w.r.t. the symmetry group of the square o r its suitably c hosen subgroup. f 00 f 01 f 10 f 11 Figure 2: Square K 2 . f 000 f 001 f 010 f 011 f 100 f 101 f 110 f 111 Figure 3: Cub e K 3 . A few other requiremen t s w ere giv en in [1], in part icular the form ula (2) in volv ed p ar ame ters atta c hed to the edges of the square. The second requiremen t of symmetry is o b viously v ery imp ortan t for the f o rm u- lation of the condition of 3-dimensional consistency of (2) . Name ly , supp ose w e hav e an elemen ta ry cub e (3 d -cell of Z 3 , cf. Fig. 3) a nd imp o se (2) to hold on three “init ia l 2 d -faces” { x 1 = 0 } : Q ( f 000 , f 010 , f 001 , f 011 ) = 0; { x 2 = 0 } : Q ( f 000 , f 100 , f 001 , f 101 ) = 0; { x 3 = 0 } : Q ( f 000 , f 100 , f 010 , f 110 ) = 0 (these are used to find f 011 , f 101 , f 110 from f 000 , f 100 , f 010 , f 001 ). Then we imp o se (2) to hold on t he ot her three “ final 2 d - faces” { x 1 = 1 } : Q ( f 100 , f 110 , f 101 , f 111 ) = 0; { x 2 = 1 } : Q ( f 010 , f 110 , f 011 , f 111 ) = 0; { x 3 = 1 } : Q ( f 001 , f 101 , f 011 , f 111 ) = 0; so for the la st field v ariable f 111 w e can find 3 (apriori) differen t rational expressions in terms of the “initial data” f 000 , f 100 , f 010 , f 001 . The 3 d -c ons i s tency is the requiremen t that these three expressions of f 111 in terms of the initial data should b e identically equal. The subtle p oint of this pro cess consists in the non-uniqueness of the mappings of a giv en square (Fig. 2) onto the six 2 d - faces of the cub e. The requiremen t of symmetry giv en ab ov e guarantee s that w e can choose an y iden t ification of the vertic es of the 2 -dimensional faces of the 3-dimensional elemen tary cub e (Fig . 3 ) with the vertice s of the “standard” square where (2 ) is g iven; certainly this iden t ification should preserv e the combinatorial structure of the square (neighbouring v ertices remain neigh b ouring). In [1] a complete classification of 3 d -consisten t 2 d -fa ce form ulas (in a sligh tly differen t setting) was obtained; in [2] a similar classification was 4 giv en f or the case when one do es no t assume that t he formula (2) is the same on all the 6 faces of the 3 -dimensional cub e. In the next sections w e giv e a symmetry classification of a ll p ossible 3 d -face formulas defined o n some “standard” 3- dimensional cub e: Q ( f 000 , f 100 , f 010 , f 001 , f 110 , f 101 , f 011 , f 111 ) = 0 (3) with resp ect to the complete symmetry group of the cub e. Here, as ev erywhere in the pa p er, indices of the field v ariables f ij k giv e the co ordinates of the corresp onding v ertices of the standard 3 d -cub e where our form ula (3) is defined. The requiremen t o f consistency is no w formulated similarly to the 2 d - case: given a 4 d -cub e with field v alues f ij k l , i, j, k , l ∈ { 0 , 1 } , one should imp ose the formula (3) on ev ery 3 d -fa ce o f it, by fixing one of the indices i, j, k , l , and making it 0 for the fa ces whic h w e will call b elow “ init ia l faces”, or resp ective ly 1 for the faces whic h w e will call “final faces”. One also needs to fix some mapping from the initial “standard” cub e (with the v ertices lab elled f ij k ) onto ev ery one o f the eigh t 3 d -faces (for example { f i 1 kl } on { x 2 = 1 } ). This can certainly b e done using the trivial lexicographic corresp ondence of the t yp e f ij k 7→ f i 1 j k . Geometrically this lexicographic correspondence is less natural since it is not inv aria n t w.r.t. the symmetry group of a 4 d -cub e. On the other hand there is an imp ortant example of suc h a non-symmetric f o rm ula corresp onding to the discrete BKP equation ([1], equation (7 6)). Another p ossibilit y t o a v oid this problem is to imp ose the requireme nt of symme try . More precisely , if one applies an y one of the transformations from the group of symmetries o f the 3 d -cub e, ( 3) shall b e transformed in to an equation with the left hand side pr op o rtion a l to the original expression Q : Q 7→ λ · Q . Since this symmetry gr o up is generated by reflections, one has λ 2 = 1, so this prop ort io nalit y m ultiplier λ may b e either (+1 ) or ( − 1) for an y particular transformation in the complete symmetry g r oup. F rom results in [1] w e kno w that there are imp ortan t 4 d -consisten t 3 d -face form ulas whic h are preserv ed under a suitable sub gr oup of the complete symmetry group of the 3 d -cub e. No classifiaction of 3 d -face formulas with suc h restricted symmetry prop erty has b eet caried out yet. 3 Symmetry clas s ification Ev ery n -dimensional face form ula Q n = 0 whic h satisfies the requiremen t of quasilin- earit y has a left hand side of the form Q n = X D q D Y i s =0 , 1  f i 1 ...i n  D i 1 ...i n (4) with constan t co efficien ts q D , where the summation is tak en ov er all 2 2 n man y 2 n - tuples D = ( D 00 ... 0 , . . . , D 11 ... 1 ), each p ow er D i 1 ...i n of the resp ectiv e ve rtex v ariable f i 1 ...i n b eing either 0 or 1. In other w ords: the 2 n indices o f q D are the exp onents o f the 2 n v ertex field v ariables f i 1 ...i n , eac h expo nen t D i 1 ...i n b eing 0 or 1. F or example, 5 Q 2 = q 1111 f 00 f 10 f 01 f 11 + q 1110 f 00 f 10 f 01 + q 1101 f 00 f 10 f 11 + . . . + q 0000 has 2 2 2 = 1 6 terms, Q 3 has resp ectiv ely 2 2 3 = 256 terms, and Q 4 has already 2 2 4 = 6 5 536 terms. In this Section w e classify n -dimensional quasilinear equations Q n = 0 for n = 2 , 3 , 4 that are in v ariant w.r.t. the complete symmetry gro up of the r esp ectiv e n -dimensional cub e. This problem can b e reduced to the enume ratio n of irreducible represen tations of this group in the space of p olynomials of the form (4). Here t his is do ne in a straigh tforward w ay: the gr oup in question is generated by one reflection w.r.t the plane x 1 = 1 / 2 and ( n − 1) diagona l reflections w.r.t. the planes x 1 = x s , s = 2 , . . . , n (here x k denote the co ordinates in R n ). T o ev ery reflection R from this generating set w e assign ( − ) or (+ ) a nd require the equalit y Q ( f ) = − Q ( R ( f )) ( resp ectiv ely Q ( f ) = Q ( R ( f )) ) to hold iden tically in all v ertex v ariables f ; this give s us a set of equations for t he co efficien ts q D . Running through all p o ssible c hoices of t he signs for the generating reflections w e solv e the united sets of simple linear equations for the co efficien ts q D for eve ry such c hoice. The main problem consists in the size of the resulting set of equations: for n = 3 w e ha v e for each combination of ± f or the 3 generating reflections around 770 equation f o r the 2 56 co efficien ts q D 1 ...D 8 ; for n = 4 ev ery set o f equations for the co efficien ts of Q 4 has around 250,000 equations for t he 65536 co efficien ts q D 1 ...D 16 . Naturally , not ev ery com bination of signs for the generating reflections is p ossible, most of the resulting sets of equations for q D 1 ...D (2 n ) allo w o nly trivial solution q D 1 ...D (2 n ) = 0. The results of our computation are giv en in table f orm in Theorem 1 b elow. 1 As it turns out, for n = 2 t hree symmetry t yp es of quasilinear expres sions Q 2 are p ossible. In the notatio n of T able 1 the first sign refers to the reflection on the line x 1 = 1 / 2 , the next sign stands for the reflection on the line x 1 = x 2 . F or example, the expressions of the first type (+ − ) ar e inv arian t w.r.t. to the reflection on the line x 1 = 1 / 2 , and sho w a change of sign after the reflection on the line x 1 = x 2 . The la st case (++) consists of expressions whic h are inv a r ian t w.r.t. any elemen t of the complete group of symmetries of the square whic h corresp onds to the choice of the (+ ) signs for the tw o generating reflections o f the square w.r.t. the lines x 1 = 1 / 2 and x 1 = x 2 . F or n = 3 alongside with the completely symmetric quasilinear expressions Q 3 (the third case (+ + +) b elow ), there are t w o other cases ( − − − ) and ( − + +) of nontrivial quasilinear Q 3 . In this nota tion the first sign refers to t he reflection on the plane x 1 = 1 / 2, the next signs stand for reflections on the planes x 1 = x 2 , x 1 = x 3 (and x 1 = x 4 for n = 4). The resulting num b ers o f free co efficien ts q D in the symmetric face formulas Q n are giv en for eac h of t he non trivial cases in T able 1. W e also give the num b er of nonzero terms in Q n for each case. Esp ecially remark able is the tot a lly sk ew-symmetric case ( − − − ) for n = 3: it has 1 T o solve the sparse but rather extensive linea r systems for the co e fficient s q D app earing a f- ter the splitting w.r.t. the v a riables f i 1 ...i n , a sp ecial linear equa tion solver had to b e w r itten. It can b e downloaded together with other mater ial related to this publication (for details see http:/ /lie. math.brocku.ca/twolf/papers/TsWo2007/readme ). 6 only one (up to a constan t m ultiple) nontrivial expression Q ( −−− ) = ( f 100 − f 001 )( f 010 − f 111 )( f 101 − f 110 )( f 011 − f 001 ) − ( f 001 − f 010 )( f 111 − f 100 )( f 000 − f 101 )( f 110 − f 011 ) . (5) Precisely this expression giv es the so called discrete Sch w arzian bi-Ka do m tsev-P etviash vili system (dBKP-system) — an in tegrable discrete system found in [6, 7] and studied in [1] where the fact of its 4 d -consistency w as first established. The dBK P- system has man y equiv alen t for ms a nd app ears in very differen t contexts . In addition to the kno wn geometric in terpretations and a reformulation as Y ang-Baiter system ([8]) , the dBKP-system may b e considere d as a nonlinear superp osition principle for the classical 2-dimensional Mouta rd transformations ( [5]). The expression (5 ) enjo ys an extra symmetry prop ert y: the equation Q ( −−− ) = 0 is in v ariant under the action of the S L 2 ( C ) group of fractional-linear transformations f 7− → ( a f + b )  ( c f + d ) (6) (all gr o up parameters a , b , c , d are the same fo r all the v ertices of the cub e). This is in fact a direct conseque nce of its uniqueness in this class. Since this S L 2 ( C ) action ob viously preserv es the symmetry ty p e of an expression w.r.t. the gr o up a ction o f the cub e symmetry group, it is reasonable to find the sub classes of S L 2 -in v ariant face equations in eac h symmetry class. This is giv en in the t hird column of the table. Theorem 1 The non e mpty symmetry classes of formulas Q n for fac e dimensions n = 2 , 3 , 4 ar e: n t yp es of symmetry , n um b er of parameters n um b er of parameters and terms in S L 2 -in v ariant sub cases 2 1) (+ − ): 1 param.; 4 terms 1) 1 param.; 4 terms 2) ( − + ): 3 param.; 10 terms 2) none 3) (++): 6 param.; 16 terms 3) 1 param.; 6 terms 3 1) ( −−− ): 1 param.; 24 terms 1) 1 param.; 24 terms 2) ( − + +): 13 para m.; 1 86 terms 2) none 3) (+++) : 22 param.; 256 terms 3) 3 pa r a m.; 1 14 terms 4 1) ( −−− − ): 9 4 param.; 29208 terms 1) 5 param.; 15480 terms 2) (+ − −− ): 77 param.; 26 112 t erms 2) none 3) ( − + ++): 349 param.; 60666 terms 3) 3 param.; 15809 terms 4) (++++ ): 402 para m.; 2 16 terms 4) 18 param.; 96314 terms T able 1: Symmetry classification of the face form ulas w.r.t the complete symmetry group of the cub e 7 In pa rticular, the explicit expression for the (no n- S L 2 -symmetric) 2 d -face form ulas are: (+ − ) : Q = q 1 ( f 11 f 10 − f 11 f 01 − f 10 f 00 + f 01 f 00 ) = q 1 ( f 11 − f 00 )( f 10 − f 01 ) , (7) ( − +) : Q = q 1 ( f 11 − f 10 − f 01 + f 00 ) + q 2 ( f 11 f 00 − f 10 f 01 ) + (8) q 3 ( f 11 f 10 f 01 − f 11 f 10 f 00 − f 11 f 01 f 00 + f 10 f 01 f 00 ) , (++) : Q = q 1 + q 2 ( f 11 + f 10 + f 01 + f 00 ) + q 3 ( f 11 f 00 + f 10 f 01 ) + (9) q 4 ( f 11 f 10 + f 11 f 01 + f 10 f 00 + f 01 f 00 ) + q 5 ( f 11 f 10 f 01 + f 11 f 10 f 00 + f 11 f 01 f 00 + f 10 f 01 f 00 ) + q 6 f 11 f 10 f 01 f 00 . The explicit fo rm of the S L 2 -symmetric face form ula for n = 2 in the first (+ − ) case is the same: Q = q 1 ( f 11 − f 00 )( f 10 − f 01 ) . F or n = 2 and the symmetric case (++) the S L 2 -symmetric fa ce formula is: Q = q 1 ( f 11 f 10 + f 11 f 01 − 2 f 11 f 00 − 2 f 10 f 01 + f 10 f 00 + f 01 f 00 ) . It should b e noted t ha t (8 ) is 3 d -consisten t; it can b e simplified using S L 2 trans- formations a nd put into explicitly linear form Q = f 11 − f 10 − f 01 + f 00 or into Q = f 11 f 00 − f 10 f 01 , so the resp ectiv e equation Q = 0 is equiv alen t to a linear equation log f 11 + lo g f 00 = log f 10 + lo g f 01 . 4 Nonexiste nce of no n trivial integrable face form u- las in other symmetry c l asses for n = 3 In this section w e give a computational pro of of our main result: Theorem 2 Among the thr e e p o s sible symmetry typ es of 3 -dim e nsional quasi l i n e ar fac e formulas given in the se c ond c olumn of T able 1, only formula (5) giv es a non- trivial 4 d -c omp atible fac e e quation. Any 4 d -c omp atible fac e form ula in the other two symmetry typ es may b e tr ansfo rm e d using the action of the gr oup S L 2 ( C ) on the field variables to one of the fol lowing line ariza b le forms: Q (1) = f 000 f 001 f 010 f 011 f 100 f 101 f 110 f 111 − σ , (10) Q (2) = f 001 f 010 f 100 f 111 − σ f 000 f 011 f 101 f 110 , (11) Q (3) = ( f 001 + f 010 + f 100 + f 111 ) − σ ( f 000 + f 011 + f 101 + f 110 ) , (12) wher e σ = ± 1 . T ec hnically , in order to find 4 d - consisten t 3 d -face formu las a mo ng t he other tw o 3- dimensional cases ( − ++) and (+++) (as listed in T able 1), one shall run the following algorithmic steps: 8 Step 1 . T ake a cop y of this Q 3 form ula, map it on to the f o ur initial fa ces of the 4 d - cub e (where one of the co ordinates x i = 0), solve the mapp ed equations with resp ect to f 0111 , f 1011 , f 1101 and f 1110 lea ving the other v ar iables free. Step 2 . Then substitute the o btained r ational expressions for f 0111 , f 1011 , f 1101 and f 1110 in to t he copies of the f ace formula mapp ed on to the four final faces (where one of the co ordinates x i = 1), finding resp ectiv ely four differen t expressions for the last v ertex field f 1111 . Step 3 . Equate these 4 expressions for f 1111 obtaining three ra tional equations in terms of 11 free v ariables f 0000 , f 0010 , f 0101 , . . . and the parametric co efficien ts q D 1 ...D 8 left free in the c hosen symmetry class. Step 4 . Remo ving the common denominato rs of the equations and splitting the resulting p olynomials w.r.t. the 11 fr ee v aria bles f ij k l one obtains a p olynomial system of equations for the free co efficien ts q D 1 ...D 8 . Step 5 . The la tter should b e solv ed, r esulting in a complete classification of 4 d - consisten t quasilinear scalar 3 d -fa ce form ulas. This a pproac h, applied in a straigh tforward w ay , results in extremely h ug e expres- sions. Ev en building the rational expressions in Step 4 in a straigh tforward w ay seems to b e unrealistic: for a typical 3 d -face formula from T a ble 1 Step 4 should end up (as our test runs allow ed for an estimate) in an expression with around 10 14 terms, whic h is b ey o nd the reac h of computer algebra systems in the foreseeable future. Eve n brute force testing of 4 d -consistency of the smallest solution (face form ula (5) whic h has no free para metric co efficien ts q D 1 ...D 8 ) results in ≈ 2 · 10 8 terms (after substitut- ing the expressions for f 0111 , f 1011 , f 1101 , f 1110 , collecting the terms ov er the common denominator and expanding the brac k ets b efore the cancellation can start in Step 4 2 ). T ec hnically this is explained by the presence of 4 different sym b olic denominators of the r a tional express ions for f 0111 , f 1011 , f 1101 , f 1110 and their v arious pro ducts. A care- ful step-b y-step substitution and cancellation o f lik e terms in sev eral stages still can b e done ev en o n a mo dest computer f o r this ( − − − ) c ase . Using the system F orm (this system w as sp ecially designed for large sym b olic computations), o ne can prov e that a ll terms finally cancel o ut for the case of the in tegrable 3 d -face formu la (5) th us giving a computational pro of of its 4 d - consistency in 3 min CPU time ( 3 GHz Intel running Lin ux SUSE 9.3 ) and less than 200 Mb disk space f or temp o rary data storage. 3 As the estimates give n in App endix A sho w, the straightforw ard appro a c h based on Steps 1 – 4 is unrealistic for the other tw o 3-dimensional cases listed in T able 1, ev en at the stage of generation of t he consistency conditions (Step 4) . In order to classify discrete in tegrable 3 d -fa ce formulas Q 3 = 0 for the case ( − ++) and the ha rdest case ( + ++) w e used a to tally different randomized “probing” strategy , explained in detail in App endices B, C, D. After the computatio n (cf. App endices B–E) the list o f candidate face fo rm ulas Q 3 for the case (++ +) included 5 form ulas (b efore the ve rification that these form ulas, 2 The commen ted F orm logfile fo r this estimate can be downloaded fro m http:/ /lie. math.brocku.ca/twolf/papers/TsWo2007/BruteForceCheck/Case---/ 3 The F orm code of this run and its logfile ca n be downloaded from http:/ /lie. math.brocku.ca/twolf/papers/TsWo2007/BruteForceCheck/Case---/ ). 9 obtained by our “ pro bing” metho d, really giv e 4 d -consisten t 3 d -form ulas). F or the case ( − ++) the list of candidate face form ulas included 3 formulas . All of them include a few f r ee pa rameters. As one can sho w, all these formulas can b e greatly simplified using the action of the group S L 2 ( C ) on the field v a riables f , r esulting in the 4 d -consisten t 3 d -face form ulas (10), (1 1), (12). The first tw o formulas (10), (11) can b e linearized using the logarithmic substitution ˜ f ij k = lo g f ij k . The expressions fo r the aforemen tioned candidates, the F orm pro cedures and their logfiles show ing the simplification pro cess can b e dow nloaded fro m http://lie. math.brocku.ca/twolf/papers/TsWo2007/ . Here w e j ust give one ex- ample of suc h a S L 2 -simplification: the Q -expression for the case Q 1 in http://lie. math.brocku.ca/twolf/papers/TsWo2007/SL2-simplification/Case+++/ is Q = q 105  f 001 f 010 f 100 f 111 + f 000 f 011 f 101 f 110  + q 107  f 001 f 010 f 100 f 110 f 111 + f 001 f 010 f 100 f 101 f 111 + f 001 f 010 f 011 f 100 f 111 + f 000 f 011 f 101 f 110 f 111 + f 000 f 011 f 100 f 101 f 110 + f 000 f 010 f 011 f 101 f 110 + f 000 f 001 f 011 f 101 f 110 + f 000 f 001 f 010 f 100 f 111  + q 2 107 q 105  f 001 f 010 f 100 f 101 f 110 f 111 + f 001 f 010 f 011 f 100 f 110 f 111 + f 001 f 010 f 011 f 100 f 101 f 111 + f 000 f 011 f 100 f 101 f 110 f 111 + f 000 f 010 f 011 f 101 f 110 f 111 + f 000 f 010 f 011 f 100 f 101 f 110 + f 000 f 001 f 011 f 101 f 110 f 111 + f 000 f 001 f 011 f 100 f 101 f 110 + f 000 f 001 f 010 f 100 f 110 f 111 + f 000 f 001 f 010 f 100 f 101 f 111 + f 000 f 001 f 010 f 011 f 101 f 110 + f 000 f 001 f 010 f 011 f 100 f 111  + q 3 107 q 2 105  f 001 f 010 f 011 f 100 f 101 f 110 f 111 + f 000 f 010 f 011 f 100 f 101 f 110 f 111 + f 000 f 001 f 011 f 100 f 101 f 110 f 111 + f 000 f 001 f 010 f 100 f 101 f 110 f 111 + f 000 f 001 f 010 f 011 f 101 f 110 f 111 + f 000 f 001 f 010 f 011 f 100 f 110 f 111 + f 000 f 001 f 010 f 011 f 100 f 101 f 111 + f 000 f 001 f 010 f 011 f 100 f 101 f 110  + 2 q 4 107 q 3 105  f 000 f 001 f 010 f 011 f 100 f 101 f 110 f 111  The simplifying transformation consists in the following steps: 1. f ij k 7→ q 105 q 107 f ij k and Q 7→ q 4 107 q 5 105 Q (this eliminates the para metric q 105 and q 107 ) 2. f ij k 7→ 1 f ij k (and remov ing the denominator in Q afterwards) 3. f ij k 7→ f ij k − 1. This pro duces the simplified form Q = f 001 f 010 f 100 f 111 + f 000 f 011 f 101 f 110 . 10 Ac kn o wledgments F or t his work facilities of the Shared Hierarc hical Academic Researc h Computing Net- w ork ( SHARCNET : www.sharcnet .ca ) w ere used. TW thanks the Konra d Zuse Institut at F reie Univ ersit¨ at Berlin and the T ec hnisc he Univ ersit¨ at Berlin where part of the w o r k was done. App endix A: The Size of Consistenc y Condi t ions The following considerations are made under the assumption of generic unkno wn co ef- ficien ts q D in the face form ula ( 4 ) not satisfying additional symmetry conditio ns. An y ( n + 1 )-dimensional h yp ercub e built from 2 n +1 v ertices f i 1 ...i n +1 , i k ∈ { 0 , 1 } has 2( n + 1) faces lo cated in the (log ical) planes x k = 0 and x k = 1, k = 1 , . . . , ( n + 1). The face relations f or the n + 1 fa ces that corresp ond to x k = 0 are 0 = X D q D Y i s =0 , 1  f i 1 ...i k − 1 0 i k +1 ...i n +1  D i 1 ...i k − 1 i k +1 ...i ( n +1) . (13) They can b e used to determine f 1 .. 101 .. 1 with the 0 b eing in the k th index p osition. Eac h of these face relations in v olv es 2 n f -v ar ia bles and thus 2 2 n terms, half of them include f 1 .. 101 .. 1 as a factor and the other half not. Solving t he face relation x k = 0: 0 = A k f 1 .. 101 .. 1 + B k (14) where A k , B k are expressions in q D , f β for f 1 .. 101 .. 1 and substituting f 1 .. 101 .. 1 = − B k / A k in an y express ion that inv olv es f 1 .. 101 .. 1 linearly ( like ot her face relatio ns) and ta king the n umerator o v er the common denominator amoun ts to m ultiplying all terms that in v olv e f 1 .. 101 .. 1 b y − B k and all other terms by A k . As A k and B k in v olv e eac h 2 2 n / 2 = 2 2 n − 1 terms this means that a substitution of f 1 .. 101 .. 1 increases the n um b er of terms b y a factor of 2 2 n − 1 , b efore cancellations and reductions will b e made. The 2 nd half of face relations for the n + 1 faces that corresp ond to x k = 1 are 0 = X D q D Y i s =0 , 1  f i 1 ...i k − 1 1 i k +1 ...i n +1  D i 1 ...i k − 1 i k +1 ...i ( n +1) . (15) Eac h one of them in v olv es f 11 .. 1 and n of those f -v a riables whic h hav e exactly one 0 as index in any one of the n + 1 index p o sitions apart fro m the k th p osition. Replacing eac h one of these n f -v aria bles b y using the corresp onding x l = 0 face relation increases the n um b er of terms b y a factor 2 2 n − 1 eac h time, giving in tot a l 2 2 n  2 2 n − 1  n = 2 2 n ( n +1) − n terms. In each substitution the degree of the co efficien ts q D increases b y one, reac hing finally n + 1. Solving o ne of the n + 1 man y x k = 1 face relations 0 = G k f 11 .. 1 + H k (16) 11 dimension of face n 2 3 4 5 # of f -v ariables i n face for mula 2 n 4 8 16 3 2 # of terms in face formula (= # of undetermined coefficients q D in Q n ) 2 2 n 16 256 65536 4 . 3 × 10 9 # of all f -v ar iables in ( n + 1)-dim. h yp ercube 2 n +1 8 16 32 6 4 # of indep. f -v ariables in ( n + 1)-dim. h yp ercube 2 n +1 − n − 2 4 11 26 57 # of n -dim. faces in ( n + 1)-dim. h yp ercube 2( n + 1) 6 8 10 12 # of consistency conditions n 2 3 4 5 upper bound on the # of terms of eac h condition 2 { 2 n +1 ( n +1) − 2 n − 1 } 5 . 2 × 10 5 1 . 4 × 10 17 2 . 8 × 10 45 1 . 9 × 10 112 total degree of the q D in eac h condition 2 n + 2 6 8 10 12 upper bound estimate of the # of equations resulting fr om splitting each condition 2 n (2 n + 2) ` 2 n +1 − n − 3 ´ 864 6 . 4 × 10 9 8 . 0 × 10 25 2 . 7 × 10 61 estimated av erage # of terms in eac h equation 2 2 n +1 ( n +1) − 2 n − 1 2 n (2 n +2) ` 2 n +1 − n − 3 ´ 606 2 . 2 × 10 7 3 . 5 × 10 19 7 × 10 50 T able 2: Some statistics of faces and consistency conditions for f 11 .. 1 and substituting f 11 .. 1 = − H k /G k in the other face relatio ns giv es n indep en- den t consistency conditions G j H k = G k H j , j = 1 , . . . , k − 1 , k + 1 , . . . , n + 1 (17) with eac h G i and H i ha ving 2 2 n ( n +1) − n / 2 terms, i.e. each consistency condition inv olving 2  2 2 n ( n +1) − n − 1  2 terms. The total n umber of terms of the n consistency conditions is th us n 2 { 2 n +1 ( n +1) − 2 n − 1 } . T o compute a n upp er b ound of the n um b er of conditions that result f rom splitting eac h consistency condition with resp ect to the indep enden t f -v ariables w e note that their hig hest degree is equal t o the to t a l degree of a ll q D , i.e. it is 2 n + 2. The only exception is f 00 ... 0 whic h do es not o ccur in t he 0 = Q n | 1 → k face relations and en ters only thro ug h substitutions, so its highest degree in the constrain ts is 2 n . W e th us get for an upp er b ound of the n um b er o f different products of differen t p o w ers of f 00 ... 0 and the other 2 n +1 − n − 3 indep enden t f - v ariables the v alue 2 n (2 n + 2)  2 n +1 − n − 3  . With this n um b er and the num b er of terms of each constraint w e get with their quotien t an estimate of the a v erage n um b er of terms in each equation ( see T able 2). R emark. Although, strictly sp eaking, these are upp er b ounds for the size of condi- tions, one shall k eep in mind that an y computer algebra system shal l gener a te al l these terms expanding the bra ck ets and o nly after generating all of them or a pa r t of them it can searc h f o r p ossible cancellations or reductions. As o ur test runs with F orm had shown, for the 3- dimensional case ( + + +) giv en in T able 1 (with a m uc h smaller n um b er of indep enden t q D than 2 2 n but the same n um b er of terms in Q 3 ), the tot al n um b er of terms to b e g enerated for eac h of the 3 consistency conditions is aro und 10 14 (compared to 1 . 4 · 10 17 in T able 2 ). A t ypical single equation for the co efficien ts q D resulting f rom splitting a partially formed consistency condition has a few thousand terms of degree 8 in the para metric q D . 12 App endix B: The computational prob lem As outlined b efore (cf. Section 4), the ta sk consists of the following steps. 1. F orm ulate the relations for the n + 1 faces x k = 0. 2. Solve them for f 11 ... 101 ... 1 . 3. F orm ulate the relations for the n + 1 faces x k = 1. 4. Perform the substitutions obtained under 2. in the relations o f 3. 5. Solve one o f the resulting r elations fo r f 11 ... 1 and 6. Substitute f 11 ... 1 in a ll other n face relations of 4. 7. Split these consistency conditions with resp ect to all the o ccurring indep enden t f -v ar ia bles to obtain an ov erdetermined system of equations fo r the unkno wns q D . 8. Find the general solution of this system. 9. Reduce the n um b er o f free parameters of the solutions using S L 2 ( C )-transformations (6). Although the algebraic system for the unkno wn co efficien t s q D is hea vily ov erdeter- mined the follow ing difficulties app ear. 1. Strictly sp eaking, in order to f orm ulate ev en only the smallest subset of conditions one would ha v e to formulate at least one consistency condition (b y p erforming steps 1., 2. fully and 3.- 6 . for at least tw o x k = 1 fa ce relations (15) b efore step 7) i.e. to generate an expression with 2 { 2 n +1 ( n +1) − 2 n − 1 } terms. 2. If one found a w ay around this hurdle then the resulting equations a re of high degree 2 n + 2 with on a v erage many t erms. 3. Eve n if one w ere able to generate 100,000’s of equations and th us find shorter ones whic h one could solv e for some unkno wns in terms of others, one would face the pro blem that man y cases and sub-sub-cases ha v e to b e inv estigated due to the high degree of the equations and 4. that one has to g enerate billions of equations t o find some that are independen t of the o nes generated so far. As w e explain b elo w, o ne can not hop e that the first, say , 10 6 conditions will b e equiv alent to t he full system of equations for q D , ev en though we hav e only v ery few unkno wn q D ’s due to the “ t riangular” form of this h uge system a s we explain in App endix D. In the computat io ns to b e describ ed in this pap er we ha ve n = 3. The full pro blem (without cubical symme try) w ould mean to generate 3 consisten cy conditions in v olving eac h ab out 10 17 terms that split into an estimate o f 10 10 p olynomial equations eac h b eing homogeneous o f degree 8 for 256 unkno wns and in v olving on av erage o v er 10 7 terms. T o mak e progress w e introduce different solving tec hniques but also simplify the problem: 13 1. W e restrict our problem to face fo rm ulas that ob ey the full cubical symmetry (cf. Section 3). This reduces the problem to 3 cases: • case ( − − − ) with 232 iden tically v anishing q D and 24 other q D dep ending on 1 parameter, • case ( − + +) with 70 iden tically v anishing q D and 186 other q D dep ending on 13 para meters, and • case (+ + +) with no iden tically v anishing q D and all 256 q D dep ending on 22 parameters. In the third case (+ + +) – the hardest case – the fact that o nly 22 of the q D are parametric simplifies the pr o blem but the simplification is limited b ecause none of the other 234 q D needs t o v anish just b ecause of the extra cubical symmetry (see Section 3). 2. W e ease the formulation of the large consistency conditions (b efore splitting them in to smaller equations) b y replacing z of the v := 2 n +1 − n − 2 indep enden t f - v ariables by z ero, and r eplacing u o f them b y random integer v a lues u n-equal zero, lea ving s := v − z − n of them in s ym b olic form. As a consequenc e, w e only get a set of necessary and not sufficien t equations for the q D but w e can rep eat this pro cedure on a gradually increasing lev el of g eneralit y ( by increasing s and lo w ering z ). A crucial feature is that the preliminary kno wledge ab out the solution is used in the for mulation o f new necessary systems of equations (see App endix C). 3. W e design a dynamic pro cess that automatically organizes an iteration pro cess that generates and solv es/simplifies/’mak es use of ’ necessary equations automat- ically . As an imp ortant feature, a set o f newly generated necessary equations is not read into the ongoing computat io n at once but gradua lly on demand (see App endix D). 4. A recen t extension is the parallelization o f differen t case inv estigations on a com- puter cluster. W e discuss p oin ts 1-3 now in more detail. App endix C: R andom probing In this App endix we explain the “probing tech nique” in more detail whic h w as men- tioned briefly in App endix B under p oint 2. in the list of tec hniques. It is based on replacing a n um b er z of the v := 2 n +1 − n − 2 indep enden t f -v ariables b y zero (for n = 3, v = 11), and replacing a n um b er u of them b y random in teger non-zero v alues, lea ving s := v − z − u of t hem in sym b olic form. The computation has 3 phases: finding solutions, v erifying the obtained solutio ns probabilistically and again rigorously . The first tw o phases use the probing tec hnique. 14 Still, it makes sense to distinguish them b ecause optimal v alues of z , u and s differ in b oth cases. In the probing t ec hnique the tw o types of replacemen ts, i.e. replacing f α b y zero or b y a no n- zero in teger, share the same disadv an tage: the n um b er of indep enden t parameters, i.e. of sym b olic f α whic h allow to split the consistency condition into man y smaller equations, is reduced by one. The adv an tage of replacing a n f α b y zero instead o f a non- zero intege r is t hat expressions shrink more. Also, replacemen ts b y non-zero in tegers often in tro duce extra solutions to the generated conditions and it app ears to b e costly t o eliminate these spurious temp o rary solutions b y leading them to a con tradiction with more conditions to b e generated based on o ther random replacemen ts. Therefore it is more pr o ductiv e to replace, for example, on e f α b y zero than to replace two f α b y non-zero integers . Consequen tly , in the first phase of fi nding solutions a t most one replacemen t b y a non-zero integer is made (i.e. u ≤ 1) and a fter starting with z = 9 , u = 0 , s = 2 one increases generality gradually b y either ch anging u from 0 to 1 and decreasing z by 1, or by decreasing u from 1 to 0 a nd increasing s b y 1 un til z = 1 , u = 0 , s = 10. W e need the o ccasional substitution b y one non-zero in teger b ecause w e w ant to increase the generalit y in as small as p ossible steps in order to a void the generation of to o ma ny to o hig h degree equations with to o ma ny terms. This w ould happ en if we decrease z b y 1, k eep u = 0 and increase s by 1. A run in full generality z = 0 , u = 0 , s = 11 is computationally pr o hibitiv e, therefore in a second phase of c onfi rming the found solution pr ob abilistic al ly one starts with z = 4 , u = 1 , s = 6 and increases generalit y b y decreasing z , increasing u and k eeping s constant un til z = 0 , u = 5 , s = 6. By testing a hypothetical solution with this final setting man y times, the correctness of the solution is confirmed with an a rbitrarily high pro babilit y . In b oth phases w e w ant to generate as few and as simple equations as p ossible in eac h generation step. So we contin ue using the same setting of z , u, s as long as p ossible (i.e. as long a s there ar e still resulting new conditions after randomly c ho osing other sets of z man y f α to b e 0, s of f α to b e k ept sym b olic and randomly assigning in teger non-zero v alues to the other u parametric f α ) b efore generalizing it, i.e. making s larger and z smaller. This is regulated by o ne pa rameter whic h specifies the maxim um n um b er of consecutiv e times that a ‘probing’ (generatio n of conditio ns) a t tempt yielded only iden t ities b efore c hanging z , u, s . In a third phase, after all hy p othetical solutions hav e b een obtained and b een c hec ked probabilistically , they a re c hec k ed a gain, now rigor ously . This has b een done either by a brute force c hec k using the computer algebra system F orm or b y using S L 2 ( C )-transformations on the field v aria bles f to reduce solutions to in tegrable trivial forms (1 0), (11), (12) in Section 4. A helpful and initially unexp ected feature of the probing tec hnique is that the resulting equations app ear to b e someho w triangular ized in the following sense . Eac h unkno wn q i 1 i 2 ...i m , m = 2 n , i j ∈ { 0 , 1 } is the co efficien t o f a pro duct of i 1 + i 2 + . . . + i m man y differen t fa ctors f α . That means, that at the b eginning of the computation when man y f α are r eplaced by zero, the q D with a high index sum do not o ccur. Only later on as fewe r f α are replaced by zero, gradually q D with higher index sum app ear in 15 the equations. On one hand t his is a go o d feature, prov iding a partially tr iangularized system of equations. On the o ther hand this means that although w e only wan t to compute a relat ively small num b er of unkno wns (for n = 3 and case (+++) these are 22 q D ) it is not enough to form ulate only a compara ble n umber of the huge total set of equations (ab out 6 . 4 × 10 9 equations for n = 3). A set of equations that is equiv alen t to the complete set of equations is only obtained tow ards the end of generalizations. F or example, q 11 ... 1 as the co efficien t of the pro duct of a ll the 2 n man y f o ccurring in at least one face form ula, at most 2 n − 1 of them b elonging to t he indep enden t f α , can only o ccur in at least one consistency condition if none of the f α o ccurring in at least one face formula is replaced b y zero, i.e. if u and s a re big enough to satisfy ( u + s ) ≥ (2 n − 1). That in turn means that millions of the early equations are redundan t which implies a large inefficiency in generating equations. This can b e a v oided by using kno wn relations q D = h D ( q ′ D ), whic h w ere deriv ed in the solution pro cess so far, as automatic simplification rules when generating new equations. Three pro blems remain to b e considered. 1. By r eplacing f α through n um b ers it ma y ha pp en that A k in any one of the x k = 0 face relations ( 14) becomes zero. Then a new equation generation attempt with differen t or more general random replacemen ts has to b e made. 2. Similarly , it ma y happ en t ha t the co efficien t of q 11 ... 1 in all n + 1 many x k = 1 face relatio ns (16) is zero. Then a differen t replacemen t has to b e tried as we ll. 3. Ev en if none of the A k in (16) b ecomes zero it may happ en tha t A k and B k in one face relation are no t prime and then a solution for t he q D whic h makes the greatest common divisor GC D ( A k , B k ) to zero is p oten tially lost when p erforming substitutions f 1 .. 101 .. 1 = − B k / A k . T he same applies to common factors of G k and H k in the single face relation (16) that is applied to replace f 11 ... 1 . Therefore eac h consistency condition has t o b e mu ltiplied with a pro duct of all common factors of an y pair A k , B k and of all common factors o f the pair G k , H k whic h is used to replace f 11 ... 1 . T o low er the computational cost one drops m ultiplicities of the factors. These factors inv olv e in general q D as w ell as f α and therefore the m ultiplication has to b e done b efore splitting the consistency condition with resp ect to the indep enden t f α . Alternativ ely one can split t he consistenc y condition b efore m ultiplication and instead m ult iply and duplicate the equations in t he fo llowing w ay . Let P ( q D , f α ) b e one of the ab o v e mentioned factors and let 0 = P b e split into a system 0 = ˆ P i ( q D ) where redundan t equations are dropp ed. 4 Instead of multiplying a constrain t 0 = C ( q D , f α ) with P , splitting the equation 0 = P C in to individual equations and factorizing all of them afterw ards, it is equiv alen t but mu ch more efficien t to split 0 = C into a system of equations 0 = ˆ C j ( q D ) and 0 = P in to a system 0 = ˆ P i ( q D ) a nd to consider the equiv alen t system 0 = ˆ P i ˆ C j , ∀ i, j . T o summarize, in order not to lo ose solutions for t he q D , the pro cedure is 4 The definition of ‘r e dundant’ dep e nds of the effor t one wan ts to spend a t this stage . In the implemen tation of this a lgorithm the p olyno mials ˆ P i are divided b y the co efficients of their leading terms with res pe ct to some ordering of the q D and then duplica te ˆ P i are dropp ed. 16 • to collect all common factors P r of all pa irs A k , B k and o f the pair G k , H k used to substitute f 11 ... 1 , • to drop duplicate factors, • to split a ll consistency conditions giving a system S of equations 0 = ˆ C j , and • to split for eac h factor P r the equation 0 = P r in to a system 0 = ˆ P r i , i = 1 . . . i r where again equations that a re redundan t within one such system are dropp ed. • If a system 0 = ˆ P r i includes a non-v anishing ˆ P r i either b ecause ˆ P r i = 1 or b ecause ˆ P r i ( q D ) is known to b e non-zero based on the inequalities that a re kno wn for some q D then this system is ignored b ecause the corresp o nding P r is non-zero. F or ev ery other suc h system 0 = ˆ P r i , replace the system S of conditions 0 = ˆ C j b y the new system ˆ S consisting of the equations 0 = ˆ P r i ˆ C j , ∀ i, j . App endix D: A Successi o n of Generatin g and Solving Equations The system o f algebraic equations for the q D is inv estigated b y the computer a lg ebra pac k age Crack that aims at solving p olynomially algebraic or differen tial systems, t ypically systems that are ov erdetermined a nd v ery large. It offers v ar ious degrees of in teractivit y from fully automatic to fully inte ractive . The pack a ge consists of ab out 4 0 mo dules whic h p erform differen t steps, lik e substitutions, factorizatio ns, shortenings, Gr¨ obner basis steps, integrations, separatio ns,... whic h can b e executed in any order. In automatic computations their application is gov erned b y a priority list where highly b eneficial, low cost and low risk (o f explo ding the size of equations) steps come first. Mo dules are tried from the b eginning of this list to its end until an attempt is success ful and then execution returns to the start of the list a nd mo dules are tried again in that order. This simple principle is refined in a n um b er of w ay s. F or more details see [11], [12]. In o rder to accommo date the dynamic generation of equations and their succes siv e use, all that ha d to b e done w as to add tw o mo dules: • one for generating a new set of necessary conditions using the ‘probing’ techniq ue from App endix C and writing the generated equations into a buffer file, and • anot her mo dule for reading one no n- trivial equation from this buffer file (i.e. for contin uously reading equations until one is obtained that is not instantly simplified t o an iden tity mo dulo the kno wn equations or un til the end of this file is reache d), and to determine the pla ce of these mo dules in the priority list. The need f or a buffer file arose b ecause the num b er of equations generated in eac h ‘probing’ is unpredictable, 17 esp ecially in view of the larg e impact that factors ˆ P r i can hav e on the num b er of equations (see the end of App endix C). 5 Some mor e misc el lane ous c o mments. As sho wn in App endix C the generated equations o ften take t he f o rm of pro ducts set equal to zero. This leads to man y case distinctions of factors b eing either zero or non-zero and consequen tly other factors b eing zero. The depth of sub-case lev els sometimes reac hes 20. Because buffer files are only v alid for the case in whic h they w ere generated (because they mak e use of case-dependen t known substitutions q D = h D ( q ′ D ) and case-dep enden t inequalities) and b ecause o f these deep lev els o f sub-cases, the n um b er o f buffer files easily reac hes 100,0 0 0 a nd more (e.g. to o man y to b e deleted with the simple UNIX command rm * ). Therefore the case lab el is enco ded in t he buffer file name allowing t o delete buffer files automatically when the case in whic h the file was created and all its sub-cases are solv ed. There is m uc h ro om for exp erimen ting with the place of the tw o new mo dules within the priority list 6 . On one ha nd one wan ts to read and create early , so tha t the ongoing computation has man y equations to c ho ose from when lo oking fo r the most suitable substitutions, shortenings, ... . The problem is that these equations are all g enerated with the same limited information on relations b et w een q D , and thu s they hav e a high redundancy . Also, dealing with many long equations do es slo w do wn Cra ck . On the other hand, giving the br a nc hing of the computation into sub-cases a higher prio rit y generates an exp onen tial gr owth of sub- and sub-sub-cases whic h drastically increases the n umber of buffer files to b e generated b ecause they are only v alid for the case they w ere generated for or fo r its sub-cases. With eac h in v estigated case, sa y q 5 = 0, the other case q 5 6 = 0 generates inequalities whic h t he pac k age Cra ck collects, up dates and mak es hea vily use of t o av oid further case distinctions as fa r as p ossible, and in this computation also to drop factors of the ˆ P mi as mentioned in App endix D. The individual cases can either b e inv estigated serially or in parallel. If t w o solutions of t w o differen t cases, for example, the solutions for q 5 = 0 and q 5 6 = 0 can b e merged in to one analytic form, if necessary b y a re-para metrization, then this is a chiev ed by one of the mo dules of Crack ([13]). App endix E: The Computation The computation was not p erformed in a single run. It fact due to t he big w orkload it extended ov er sev eral mon ths. Simple sub cases w ere solv ed initia lly whereas harder 5 This arr a ngement is similar to the design of CPU chips which have not o nly access to their own register memor y ( ∼ the e quations k nown within Crack ) and access to the hard disk ( ∼ the p ossibility to ca ll the ‘pro bing ’ mo dule) but whic h also have a ccess to ca che memo ry ( ∼ the buffer file). 6 apart from the neces sity to give the ‘reading fr om the buffer’ mo dule a higher priority than the ‘creating of a buffer’ mo dule in or der to try reading and emptying a buffer file firs t (if av ailable and not already read completely) and to create a new buffer file only if none is av ailable o r if the av ailable one is alrea dy completely rea d in 18 ones were completed only after the probing tec hnique from App endix C and its au- tomatic in terpla y with the pac k age Crack w ere dev elop ed. Eve n then it to ok some time to fine-tune parameters and put the new mo dules in the righ t place within the priorit y list of pro cedures in Crack . Finally , it w as nearly p ossible to do the com- putation fully automatically , o nly a few times the prop er case distinctions had to b e initiated man ually a t the righ t time to b e a ble to complete the computation. If one w ould add up purely the necessary computation time without runs follo wing a p o or man ual choice of case distinctions leading to computatio ns whic h generated to o large systems and whic h could not b e completed then this w o uld amount to 2 w eeks o f CPU time on a 3GHz AMD64 PC. References [1] V.E. Adler, A.I. Bob enk o, Y u.B. Suris, Class ific ation of inte gr able e quations on quad-gr aphs. The c onsi s tency appr o ach . Comm. Math. Ph ys., 2003, v. 233, p. 513– 543. [2] V.E. Adler, A.I. Bob enk o, Y u.B. Suris, Di s cr ete nonline ar hyp erb ol i c e quations. 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