On Algorithmic Estimation of Analytic Complexity for Polynomial Solutions to Hypergeometric Systems
aa r X i v : . [ c s . S C ] J u l On Algorithmic Estimation of AnalyticComplexity for Polynomial Solutions toHypergeometric Systems
V. A. Krasikov
Plekhanov Russian University of EconomicsLaboratory of Artificial IntelligenceStremyanny 36, Moscow, 115054, Russia
Abstract.
The paper deals with the analytic complexity of solutions tobivariate holonomic hypergeometric systems of the Horn type. We obtainestimates on the analytic complexity of Puiseux polynomial solutionsto the hypergeometric systems defined by zonotopes. We also proposealgorithms of the analytic complexity estimation for polynomials.
Keywords:
Hypergeometric systems of partial differential equations,holonomic rank, polynomial solutions, zonotopes, analytic complexity,differential polynomial, hypergeometry package
The notion of complexity is widely used in Mathematics and Computer Sciencein the context of several various abstract objects. The computational complexityof algorithms, the algebraic complexity of polynomials, the Rademacher complex-ity in the computational learning theory or the social complexity in the socialsystems are the concepts of great importance in the corresponding fields of sci-ence. The present work is devoted to the particular type of complexity – theanalytic complexity of bivariate holomorphic functions.The notion of analytic complexity is closely related to Hilbert’s 13th problem,which was solved by A.N. Kolmogorov and V.I. Arnold in 1957 [1]. The initialformulation of Hilbert’s 13th problem asks whether any continuous function ofseveral variables can be represented as a finite superposition of bivariate func-tions [17]. The main purpose of the theory of the analytic complexity is findingsimilar representations for analytic functions. The objects under considerationin this theory are the analytic complexity classes.
Definition 1. (See [2]). Let O ( U ( x , y )) denote the set of holomorphic func-tions in an open neighborhood U ( x , y ) of a point ( x , y ) ∈ C . The class Cl of analytic functions of analytic complexity zero is defined to comprise the func-tions that depend on at most one of the variables. A function f ( x, y ) is saidto belong to the class Cl n of functions with analytic complexity n > V. A. Krasikov exists a point ( x , y ) ∈ C and a germ f ( x, y ) ∈ O ( U ( x , y )) of this functionholomorphic at ( x , y ) such that f ( x, y ) = c ( a ( x, y ) + b ( x, y )) for some germsof holomorphic functions a, b ∈ Cl n − and c ∈ Cl . If there is no such repre-sentation for any finite n, then the function f is said to be of infinite analyticcomplexity.Example 1. A generic element of the first complexity class Cl is a function ofthe form f ( f ( x ) + f ( y )) . A function in Cl can be represented in the form f ( f ( f ( x ) + f ( y )) + f ( f ( x ) + f ( y ))) , where f i ( · ) are univariate holomor-phic functions, i = 1 , . . . , Cl n , n ∈ N there exists a system of dif-ferential polynomials with constant coefficients ∆ n which annihilates a functionif and only if it belongs to Cl n . Example 2. (See [2]). For a bivariate function f ( x, y ) consider the differentialpolynomial ∆ ( f ) = f ′ x ( f ′ y ) f ′′′ xxy − ( f ′ x ) f ′ y f ′′′ xyy + f ′′ xy ( f ′ x ) f ′′ yy − f ′′ xy ( f ′ y ) f ′′ xx . This differential polynomial vanishes if and only if its argument f ∈ Cl . The problem of defining whether a function belongs to an analytic complexityclass is equivalent to computing the corresponding differential polynomial. Notethat this is the problem of great computational complexity [4,11], thus a directapproach to its solution appears to be inappropriate.An important question is a possible connection between the classes of finiteanalytic complexity and hypergeometric functions. In this paper we considerhypergeometric functions as solutions of hypergeometric systems in the sense ofHorn [8,10]. We choose a matrix A ∈ Z m × n = ( A ij , i = 1 , . . . , m, j = 1 , . . . , n )and a vector of parameters c = ( c , . . . , c m ) ∈ C m . We denote the rows of thismatrix by A i , i = 1 , . . . , m. Definition 2.
The hypergeometric system (or
Horn system ) Horn(
A, c ) is thefollowing system of partial differential equations: x j P j ( θ ) f ( x ) = Q j ( θ ) f ( x ) , j = 1 , . . . , n, (1)where P j ( s ) = Y i : A ij > A ij − Y l ( i ) j =0 (cid:16) h A i , s i + c i + l ( i ) j (cid:17) ,Q j ( s ) = Y i : A ij < | A ij |− Y l ( i ) j =0 (cid:16) h A i , s i + c i + l ( i ) j (cid:17) , and θ = ( θ , . . . , θ n ) , θ j = x j ∂∂x j . nalytic Complexity of Solutions to Hypergeometric Systems . . . 3 Definition 3.
The system of equations Horn(
A, c ) is called nonconfluent if m P i =1 A i = 0 . It has been conjectured in [14] that any hypergeometric function has finiteanalytic complexity. Hypergeometric systems of equations differ greatly fromthe differential criteria for the analytic complexity classes, but numerous com-puter experiments suggest the hypothesis is true in a lot of particular cases [6,7].The case of hypergeometric systems with low holonomic rank has been consid-ered in [9].The set of functions of infinite analytic complexity is also a matter of interest.Until recently, all known examples of such functions were the differentially tran-scendental functions, that is, the functions that are not solutions to any nonzerodifferential polynomial with constant coefficients. Important examples of differ-entially algebraic functions of infinite analytic complexity have been presentedin [15,16].
Definition 4.
Let l i denote the generator of the sublattice { s ∈ Z n : h A i , s i =0 } and let k i be the number of elements in the set { A , . . . , A m } , which coincidewith A i . Let us define a polygon P ( A ) (see [13]) as the integer convex polygonwhose sides are translations of the vectors k i l i , the vectors A , . . . , A m being theouter normals to its sides. We will say that hypergeometric system Horn( A, c ) is defined by the polygon P ( A ) . Definition 5.
A polygon is called a zonotope if it can be represented as theMinkowski sum of segments.In this article we investigate the analytic complexity of solutions to hyperge-ometric systems of equations (1) defined by zonotopes.The present paper is organised as follows. In Section 2 we investigate par-ticular cases of hypergeometric systems defined by zonotopes and analyze theanalytic complexity of their solutions. We formulate and prove an estimate ofthe analytic complexity for polynomial solutions to such systems in terms ofthe defining matrices and parameter vectors. In Section 3 we present algorithmsfor finding the supports of polynomial solutions to hypergeometric systems andestimating the analytic complexity of polynomials. In Section 4 we consider ex-amples of hypergeometric systems and estimate the analytic complexity of theirsolutions.We use the Wolfram Mathematica package HyperGeometry for solving hyper-geometric systems we investigate in this article. The package is available for freepublic use at ,the description of available functions is given in [12].
Let us consider the special case of hypergeometric systems defined by zonotopes.Numerous experiments suggest that the analytic complexity of polynomial solu-
V. A. Krasikov tions to such systems can be much lower than its estimate based on the numberof their monomials.The set of hypergeometric systems defined by zonotopes enjoys the followingproperties:a) these systems are holonomic for the generic parameter value;b) the holonomic rank of hypergeometric systems (see Theorem 2.5 in [5]) isgiven by rank(Horn(
A, c )) = d d − X A i , A j lin. dependent ν ij , where d j = m P i = 1 A ij > A ij , j = 1 , ν ij = (cid:26) min( | A i A j | , | A j A i | ) , if A i , A j are in opposite open quadrants of Z , , otherwise.For the hypergeometric systems defined by zonotopes there is another formulafor computing their holonomic rank (see Proposition 1 in [9]), which in some casesmay be more suitable;c) the rows of the matrix defining such a system can be united into two matricesˆ A, − ˆ A ;d) for a hypergeometric system defined by a zonotope one can always chooseparameter values such that any solution to the resulting system is a polynomial(see [10]). Namely, for such a hypergeometric system Horn( A, c ) , where the ma-trix A contains 2 k rows, let α = ( α , . . . , α k ) be a part of the parameter vector c, corresponding to the matrix ˆ A (see the property (c) above) , β = ( β , . . . , β k ) be apart of this vector, corresponding to − ˆ A. Then the general solution to Horn(
A, c )is a polynomial if − α i − β i ∈ N \{ } for i = 1 , . . . , k. The simplest case of a zonotope is a parallelogram. The analytic complexityestimate of the solutions to the systems defined by parallelograms is the basisfor more complex cases.
Proposition 1.
The analytic complexity of a hypergeometric systems definedby a parallelogram cannot exceed 2 . Proof.
The solution to the hypergeometric system defined by a parallelogram,has been described in Proposition 4.7 in [10]. For the bivariate system ( n = 2)this formula leads to( x − a x − a ) α (cid:0) x − a x − a (cid:1) − α − β · ( x − a x − a ) α (cid:0) x − a x − a (cid:1) − α − β , where A − = (cid:18) a a a a (cid:19) , c = ( α , α , β , β ) . The monomials x − a x − a and x − a x − a both belong to Cl , thus for any univariate analytic functions φ ( · ) , ψ ( · )the product φ ( x − a x − a ) · ψ ( x − a x − a ) belongs to Cl . (cid:3) nalytic Complexity of Solutions to Hypergeometric Systems . . . 5 The following example shows that the solutions to hypergeometric systemsdefined by more complex polygons can be of a low analytic complexity.
Example 3. Simple zonotope.
Let us consider the hypergeometric system Horn( A ,c ) defined by the matrix A = (cid:18) − − − − (cid:19) T and the parameter vector c = ( − , , − , , − , . The holonomic rank of this system is equal to 3 . The hypergeometric system Horn( A , c ) is defined by a zonotope, since rows of A correspond to normal vectors to sides of the polygon. Representation of thiszonotope in the form of the Minkowski sum of segments is shown in Figure 1 ✻ ✲❅❅❅❅❅❅ r r rrr r = ❅❅❅ rr + rr + rr Fig. 1.
Polygon, defining the system Horn( A , c ) , and its representation as theMinkowski sum of segments The support for the system Horn( A , c ) polynomial solution is shown inFigure 2. Let us consider the part of the solution whose support is boundedby the divisors parallel to coordinate axis. This polynomial p ( x, y ) belongs tothe basis of the linear space of solutions to Horn( A , c ). Note that p ( x, y )contains 90 monomials (we do not put here the whole expression due to itslarge size) and the known estimates for polynomials imply that the analyticcomplexity of p ( x, y ) does not exceed 5 . Indeed, the support of p ( x, y ) lies inthe union of 10 lines parallel to x axis. The analytic complexity of polynomialwhose support lies on a straight line parallel to axis cannot exceed 1 . Then theanalytic complexity of the sum of k such polynomials cannot exceed 1 + ⌈ log k ⌉ . Later we prove that the analytic complexity of p ( x, y ) is actually equal to 3 . In general, appending a pair of rows ( a i , b i ) , ( − a i , − b i ) to the matrix defininga hypergeometric system is equivalent to adding a pair of parallel divisors in theexponent space. Let the hypergeometric system Horn( A , c ) be defined by aparallelogram, and p ( x, y ) = P ( s,t ) ∈ S c s,t · x s y t be a polynomial solution of thissystem, S be its support. Adding a pair of divisors in the exponent space leadsto the system with the solution given by p ( x, y ) = X ( s,t ) ∈ S Γ ( α s + β t + γ + 1)Γ ( α s + β t + γ ) · c s,t · x s y t = X ( s,t ) ∈ S ( α s + β t + γ ) x s y t V. A. Krasikov ✻ ✲ ❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅ ss ss ss ss ss ss ss ss ss ss ss s ss ss ss ss ss ss ss ss ss ss s s s s s s s s s ss s s s s s s s s s ss s s s s s s s s s ss s s s s s s s s s ss s s s s s s s s s ss s s s s s s s s s ss s s s s s s s s s ss s s s s s s s s s ss s s s s s s s s s ss s s s s s s s s s s
Fig. 2.
The support for the solution of the system Horn( A , c ) = ( α θ x + β θ y + γ ) X ( s,t ) ∈ S c s,t x s y t = ( α θ x + β θ y + γ ) p ( x, y ) . Using this formula repetitively we obtain the solution for k pairs of additionaldivisors: p k ( x, y ) = k Y j =1 ( α j θ x + β j θ y + γ j ) p ( x, y ) . Thus the estimate for the analytic complexity of p k ( x, y ) depends on theanalytic complexity of p ( x, y ) . This dependence is described in detail in the fol-lowing Proposition and its corollaries. There θ x = x ∂∂x , θ y = y ∂∂y and α i , β i , γ i ∈ C , i = 0 , , . . . Proposition 2.
Let f ( x, y ) be a Cl n function. Then( α θ x + β θ y + γ ) f ( x, y ) ∈ Cl n +1 . Proof.
The proof of the statement is based on the proof of Proposition 8 in [3].Consider the result of the differential operator α θ x + β θ y action on the function f ( x, y ) . Using the induction by n we can prove that this function belongs to Cl n . nalytic Complexity of Solutions to Hypergeometric Systems . . . 7 For n = 1 we can represent f ( x, y ) in the form f ( x, y ) = c ( a ( x ) + b ( y )) . ( α θ x + β θ y ) c ( a ( x ) + b ( y )) = c ′ ( a ( x ) + b ( y )) · ( α xa ′ ( x ) + β yb ′ ( y )) , and this function belongs to Cl as a product of Cl functions. If the statementholds for all n < N, and f ( x, y ) belongs to Cl N , which means it can be rep-resented as f ( x, y ) = h ( f ( x, y ) + f ( x, y )) , where f ( x, y ) , f ( x, y ) ∈ Cl N − , then ( α θ x + β θ y ) h ( f ( x, y ) + f ( x, y )) = h ′ ( f ( x, y ) + f ( x, y )) ·· (( α θ x + β θ y ) f ( x, y ) + ( α θ x + β θ y ) f ( x, y )) . Both of the functions f ( x, y ) and f ( x, y ) belong to Cl N − , so the estimate ofthe analytic complexity for ( α θ x + β θ y ) f i ( x, y ) , i = 1 , Cl N − . Then theirsum belongs to Cl N − and, after the multiplication of the result by h ′ ( f ( x, y )+ f ( x, y )) ∈ Cl N , the product belongs to Cl N . Thus we conclude that for any n, if f ( x, y ) ∈ Cl n then ( α θ x + β θ y ) f ( x, y ) ∈ Cl n . Adding γ f ( x, y ) ∈ Cl n tothis expression we obtain a function in Cl n +1 . (cid:3) Corollary 1.
For any f ( x, y ) ∈ Cl n the analytic complexity of k Y j =1 ( α j θ x + β j θ y + γ j ) f ( x, y )cannot exceed 2 k ( n + 1) − . Corollary 2.
Assume that the analytic complexity of a polynomial solution p ( x, y ) to the hypergeometric system Horn( A , c ) does not exceed n, S is asupport of p ( x, y ) . Let the matrix A be obtained from A by appending k pairsof vectors ( a i , b i ) , ( − a i , − b i ) , vector c be obtained from c by appending 2 k ele-ments. Then the analytic complexity of a polynomial solution with the support S to the hypergeometric system Horn( A, c ) does not exceed 2 k ( n + 1) − . While this estimate is rough when we use it for several additional pairs ofdivisors ( k > k = 1 it can be quite accurate. Example 3. (Continued).
Let us use Corollary 2 to estimate the analytic com-plexity of a solution to the system Horn( A , c ) . To do this, consider the systemHorn( ˜ A , ˜ c ) , defined by the matrix ˜ A = (cid:18) −
10 1 − (cid:19) T and the vector of pa-rameters ˜ c = ( − , − , , . This system differs from the original one only byan absence of the pair of divisors with the normal vectors (1 ,
1) and ( − , − . Thus the support of the solution to the system Horn( ˜ A , ˜ c ) coincides with thesupport of p ( x, y ) . Note that this system is defined by a parallelogram and henceby Proposition 1 the analytic complexity of its solutions cannot exceed 2 . Com-putations show that the basis in the space of solutions to the system Horn( ˜ A , ˜ c )consists only of one function: ( x − ( y − ∈ Cl , then p ( x, y ) ∈ Cl by V. A. Krasikov
Corollary 2. Supports of two other solutions to Horn( A , c ) lie on two paral-lel divisors, so a linear combination of these solutions belongs to Cl , and thegeneral solution to Horn( A , c ) is a function in Cl . The resulting analytic complexity estimate of solutions to hypergeometricsystems defined by zonotopes is formulated in the following theorem.
Theorem 1.
Let Horn(
A, c ) be a hypergeometric system and Horn(
A, c ) is de-fined by a zonotope. Assuming the matrix A contains 2 k rows, consider matri-ces ˆ A and − ˆ A such that the union of their rows coincides with the set of rowsof A. Let α be a part of the parameter vector c, corresponding to the matrixˆ A, β be a part of this vector, corresponding to − ˆ A, and define the vector ˆ c withelements ˆ c i = − α i − β i . If ˆ c i ∈ N \{ } , i = 1 , . . . , k, then the analytic complexity of the general solu-tion to Horn( A, c ) does not exceedmin (cid:18) · k − − ⌈ log k ( k − ⌉ , ⌈ log (max i ˆ c i + 1) ⌉ + ⌈ log ( k − ⌉ (cid:19) . Proof.
The condition ˆ c i ∈ N \{ } provides the existence of a polynomial basis inthe space of solutions to Horn( A, c ) . The matrix A contains 2 k rows, so supportsof the solutions are bounded by k pairs of divisors. The union of these supportsis a subset of k ( k − parallelogram intersections (it is a sum of an arithmeticprogression) and in every intersection the solution belongs to Cl · k − − (byCorollary 2).On the other hand, there is the estimate based on the number of parallel linesconnecting the points of the support (see Proposition 4 in [3]). While the analyticcomplexity of any polynomial with the support belonging to a straight line doesnot exceed 2 , the number of these lines for every pair of divisors equals ˆ c i + 1 . Thus for any pair of divisors, the part of the solution, belonging to intersectionsof this pair and any other pairs cannot exceed 2 + ⌈ log (max i ˆ c i + 1) ⌉ . Notethat there is no need to use all of k pairs of divisors to estimate the analyticcomplexity of the general solution this way, since k − k ( k − parallelogram intersections, then to find theestimate, based on pairing of k − (cid:3) Let us order ˆ c i by the ascension and then v be a vector with the elements v i =min (cid:0) ⌈ log (ˆ c i + 1) ⌉ , · k − − ⌈ log ( k − i ) ⌉ ) , i = 1 , . . . , k − . To findmore accurate value for the analytic complexity estimate from Theorem 1, onecould use Algorithm 1 from Section 3 using v as an input vector. The accuracyis obtained due to the fact that the vector v provides the decision of betterestimate for every pair of divisors, since ˆ c i may have high values not for all ofthem. nalytic Complexity of Solutions to Hypergeometric Systems . . . 9 To estimate the analytic complexity of the general solution to the hypergeometricsystem from Theorem 1 one can use the following algorithm.
Algorithm 1:
Finding the analytic complexity estimate for the sum offunctions
Input: c = { c , c , . . . , c n } - a set of known estimates of the analyticcomplexity values for bivariatefunctions f ( x, y ) , f ( x, y ) , . . . , f n ( x, y ) , where ( x, y ) ∈ C . Output: N - an estimate for the analytic complexity of the function n P i =1 f i ( x, y ) . while c contains more than 1 element do find 2 minimal elements of c, namely, c i and c j . c = ( c ∪ { max( c i , c j ) + 1 } ) \{ c i , c j } . N ← only element of c. Algorithm 1 is finite, since at each step the number of elements in c de-creases by 1 . The following algorithm allows one to find the support of a polynomial so-lution to a given hypergeometric system defined by a zonotope, provided thatsuch a solution exists. The algorithm is based on Proposition 4.7 in [10].
Algorithm 2:
Constructing the support for the polynomial solution to thehypergeometric system
Input: the matrix A, the parameter vector c for the hypergeometricsystem Horn( A, c ) defined by a zonotope
Output: supp - the support for the polynomial solution to Horn(
A, c ) . supp ← {} find ˆ A : rows( ˆ A ) ∪ rows( − ˆ A ) = rows( A ) for ( r i , r j ) ⊂ rows( ˆ A ) , i < j do A i,j ← ( r i , r j ) T α ← elements of c corresponding to ( r i , r j ) β ← elements of c corresponding to ( − r i , − r j ) if − α j − β j > for j = 1 , then supp = supp ∪ Supp (cid:18) x − A − i,j α (cid:16) x − A − i,j e (cid:17) − α − β (cid:16) x − A − i,j e (cid:17) − α − β (cid:19) else the general solution to Horn( A, c ) is not a polynomialFor some pairs of rows r i , r j solution to the corresponding system defined bya parallelogram is not a polynomial. In this case, part of the basis in the solutionspace can still consists of polynomials, and their supports can be found by themeans of Algorithm 2. The following algorithm allows one to compute the analytic complexity ofany given bivariate polynomial.
Algorithm 3:
Finding the analytic complexity estimate for the polynomial
Input: p ( x, y ) - a polynomial, x, y ∈ C . Output: N - an estimate for the analytic complexity of p ( x, y ) . result ← short ← {} polys ← { p i ( x, y ) | p ( x, y ) = P i p i ( x, y ) , Supp p i ( x, y ) || Supp p j ( x, y ) ∀ i, j } for p ∈ polys do curr = getShort ( p ) if curr short then result += 1 short = short ∪ curr N ← ⌈ Log ( result ) ⌉ The main improvement of this algorithm compared to the existing ones is itsability to distinct the powers of lower degree polynomials included in the originalpolynomial as summands. Without this feature, even the analytic complexityof the function like p ( a ( x ) + b ( y )) ∈ Cl , where p ( t ) , a ( x ) , b ( y ) are univariatepolynomials, is estimated based on its support, which becomes very complexwith the growth of degree of p ( t ) . The input of the function getShort () is a homogeneous polynomial and theoutput contains elements of its decomposition into the sum of powers. Notethat the definition of polys assumes the ambiguity of the representation of thepolynomial as the sum of finitely many polynomials with their supports lying inparallel straight lines. Any of such representations give an estimate, but some ofthem may be better than other ones.
Example 3. (Continuation).
Let us replace the parameter vector c in the sys-tem Horn( A , c ) by the vector ( k, , , , , . The corresponding system isgiven by xθ x ( θ x + θ y + k ) − θ x ( θ x + θ y ) ,yθ y ( θ x + θ y + k ) − θ y ( θ x + θ y ) . A basis in its solution space is given by 1 , log xx − + P k − j =1 ( − j j ( x − j , log yy − + P k − j =1 ( − j j ( y − j , so there is no polynomial basis for these parameter values. Never-theless, the analytic complexity of the general solution is equal to 1 . The present example shows that the analytic complexity of solutions to hy-pergeometric systems can be heavily dependent on parameter vectors definingthese systems. A resonant choice of their parameters can drastically reduce theanalytic complexity of general solutions to such systems. nalytic Complexity of Solutions to Hypergeometric Systems . . . 11
Example 4. An octagon zonotope.
Consider Example 6.8 in [10]. In order to findthe analytic complexity of a polynomial solution to the hypergeometric systemdefined by the matrix A = (cid:18) − − − − − − − − (cid:19) T and the vector of parameters c = (3 , − , − , , − , − , − , −
1) we can use thebasis of the solutions to this system, computed in the book. There are 3 solutionswhose analytic complexity equal to 2, and 28 solutions in Cl , two of them alsobelonging to Cl . Therefore the analytic complexity of the general solution tothis system cannot exceed 7 . Note that this estimate is based on a trivial pairingof the basis functions, but very specific structure of the solution support makesit possible to estimate the analytic complexity not to exceed 6 . Let us estimate the analytic complexity of the general solution to this system,using Theorem 1. The vector ˆ c, ordered by the ascension, is (1 , , , . Then thevector v = (3 , ,
4) (it includes only support-based estimates, because of lowvalues of the elements of ˆ c ) , and, by using Algorithm 1, we conclude that thegeneral solution belongs to Cl . Note that this estimate coincides with the onewe have obtained by hand.Futhermore, we can estimate the analytic complexity of a solution to anyhypergeometric system we obtain by appending a pair of rows to A (the onlycondition is that these rows are not collinear to the rows of A ). Note that thisestimate does not depend much on the difference between new parameters. Ifthis difference is great, it becomes the last element of the ordered vector ˆ c, and does not affect the new vector v, the new element of the vector v is equalto 2 + ⌈ log (3 + 1) ⌉ = 4 , and the resulting analytic complexity is 6 . On thecontrary, if this difference is low, for example, if it is equal to 1 , the new vector ˆ c =(1 , , , , , the new vector v = (3 , , , , and the analytic complexity is alsoequal to 6 . Thus we conclude that the addition of 2 rows to the matrix A doesnot affect the analytic complexity of the solution to the system. Example 5. A decagon zonotope.
Consider the hypergeometric system Horn( A , c ) , defined by the matrix A = (cid:18) − − − −
30 0 − − − − (cid:19) T (2)and the parameter vector c = ( − , , , − , , − , − , , − , . The zonotopedefining the matrix A is shown in Figure 3.The holonomic rank of the system Horn( A , c ) equals 34 . The support tothe solution of this system computed by the means of Algorithm 2 is shown inFigure 4.Polynomial basis in the solution space to Horn( A , c ) consists of the 4 mono-mials x y , x / y , x y , x / y and 30 polynomials1 xy + 5643637 xy + 2470958281 xy + 3294608281 xy + 27455286 y + 8236549 y + 74128549 y + 7248127 y , ✻ ✲✁✁❇❇❇❇❏❏❏❏✁✁❇❇❇❇❏❏❏❏ r r rrrrrrr r Fig. 3.
The zonotope which defines the matrix (2) ✻ ✲ − −
10 10 (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏ rrrrr rrrrr r r r rr r r r rrrr r r r rrrrrrrrrrrrrrrrrrrrr rr rrrrrrrrrrrrr rrrrrrrrr rrrrrrrrr rrrrrrrrrrrrr r r r r rr r r r rr r r r rr r r r r
Fig. 4.
The support for the solution of the system Horn( A , c )nalytic Complexity of Solutions to Hypergeometric Systems . . . 13 y x − y x + 4 y − y , y / x − y / x + 1323 y / x − y / + √ y, − y x − y x − y x − y x + 3 y y
26 + 36 y
143 + y , xy − xy , y x + 4 y x + 50 y
81 + y , x / y − x / y x / y − x / y , x y − x y − x y x y , y x / + 84656 y x / + y x / , x / y + 451 x / y , y x / + 5220 y x / + 36575 y x / + y x / , y x + 33 y x + y x , x / y + 1378 x / y , − x / y + x / y + 119286 x / y , − x / y + x / y − √ xy , x y + x, x / y / + 14382 x / y / + x / y / , x / y / + 345 x / y / + x / y / , x / y + 261 x / y , x / y / x / y / + x / y / , x / y / + 731 x / √ y + 1763 x / √ y ,x / y / + 32680 x / y / + 1558261 x / √ y, x / y / + x / y / + 65 x / y / , −
166 5 x y + 57 x y − x y + x , x / y / − x / y / x / y / ,x / y / − x / y / x / y / , x / y / − x / y / − x / y / ,x / y / + 5824 x / y / − x / y / , x y + 8 x y − x y − x y − x y ,x / y + 828 x / y − x / y + 21758 x / y − x / y . There are 14 functions in Cl and 20 functions in Cl \ Cl among these polyno-mials.The analytic complexity estimate of the general solution obtained by thepairing of these functions is Cl . Theorem 1 gives the following estimate: ˆ c =(2 , , , , , v = (4 , , , , then the general solution belongs to Cl . The following examples present hypergeometric systems defined by non-zonotopepolygons, with solutions having low analytic complexity.
Example 6. A pentagon.
The matrix (cid:18) − − − − (cid:19) T and the vector ofparameters ( − , , , − , − , − , −
2) define the hypergeometric system x ( θ x + θ y − θ x − − θ x ( θ x − ,y ( θ x + θ y − θ y − − θ y ( θ y − . (3)This system is holonomic and its holonomic rank equals 4. The pure basis(see [10]) in its solution space is given by the Taylor polynomials x y , − x − y +12 xy, x − x + x − x y +4 x y, y − xy − y +4 xy + y . The first and the second of these polynomials belong to Cl , the third and thefourth belong to Cl . Thus the general solution is a function in Cl . ✻ ✲ st ❅❅❅❅❅❅ rr rr ❞❞ ❞❞ ❞t tt tt ✈ ✻ ✲❅ rrr rrr rr Fig. 5. a): the supports of solutions to the system (3); b) polygon defining the sys-tem (3)
Example 7. A trapezoid, high holonomic rank.
The Ore-Sato coefficient ϕ ( s, t ) =Γ( s + t )Γ( s ) k − Γ( − s ) k Γ( − t ) defines the following hypergeometric system withholonomic rank k : xθ k − x ( θ x + θ y ) − ( − k θ kx ,y ( θ x + θ y ) + θ y . A basis in its solution space is given by { log j (( y + 1) /x ) , j = 0 , . . . , k − } . The generating solution equals log k − (( y + 1) /x ) . Thus the general solution tothis system belongs to Cl by the conservation principle. This example showsthat the analytic complexity of solutions to hypergeometric systems with highholonomic rank can still be low. Example 8. A triangle with no symmetries.
The hypergeometric system x ( θ x + θ y − θ x + 2 θ y − − (2 θ x + 3 θ y − θ x + 3 θ y − ,y ( θ x + θ y − θ x + 2 θ y − θ x + 2 θ y − − (2 θ x + 3 θ y − θ x + 3 θ y − θ x + 3 θ y −
6) (4)is holonomic and its holonomic rank equals 6. The pure basis in its solution spaceis given by the Laurent polynomials x − y , x − y , x y − , x y − , y + 2 x − y , nalytic Complexity of Solutions to Hypergeometric Systems . . . 15 ✻ ✲ st ◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗ ◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗ ◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗ ❅❅❅❅❅❅❅❅❅❅❅❅ ❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍ ❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍ r r r r❜ ❜ t tt t tt tt t Fig. 6.
The supports of solutions to the system (4) x + 12 x + x + 4 x y − + 6 x y − − x y − − x y − − xy − x y. Here the small filled circles correspond to monomial solutions, the two emptycircles indicate the binomial solution and the big filled circles correspond to theremaining polynomial solution. The analytic complexity of the general solutionto the system (4) does not exceed 5 . Acknowledgements.
This research was performed in the framework of the statetask in the field of scientific activity of the Ministry of Science and Higher Edu-cation of the Russian Federation, grant no. FSSW-2020-0008.
References
1. Arnold, V.I.: On the representation of continuous functions of three variables bysuperpositions of continuous functions of two variables. Mat. Sb. (1), 3–74 (1959)2. Beloshapka, V.K.: Analytic complexity of functions of two variables. Russian J.Math. Phys. (3), 243–249 (2007)3. Beloshapka, V.K.: Analytical complexity: Development of the topic. Russian J.Math. Phys. (4), 428–439 (2012)4. Beloshapka, V.K.: On the complexity of differential algebraic definition for classesof analytic complexity, Math. Notes, (3), 323–331 (2019)5. Dickenstein, A., Matusevich, L.F., Sadykov, T.M.: Bivariate hypergeometric D-Modules. Advances in Mathematics , 78–123 (2005)6. Dickenstein, A., Sadykov, T.M.: Algebraicity of solutions to the Mellin system andits monodromy. Dokl. Math. (1), 80–82 (2007)7. Dickenstein, A., Sadykov, T.M.: Bases in the solution space of the Mellin system.Sbornik Mathematics, (9), 59–80 (2007)8. Horn J.: ¨Uber die Konvergenz der hypergeometrischen Reihen zweier und dreierVer¨anderlichen. Math. Ann. , 544–600 (1889)6 V. A. Krasikov9. Krasikov, V.A.: Analytic Complexity of Hypergeometric Functions Satisfying Sys-tems with Holonomic Rank Two. Lecture Notes in Computer Science, , 330–342(2019)10. Sadykov, T.M., Tanabe, S: Maximally reducible monodromy of bivariate hyperge-ometric systems. Izv.: Math. (1), 221–262 (2016)11. Sadykov, T.M.: Beyond the First Class of Analytic Complexity. Lecture Notes inComputer Science, , 335–344 (2018)12. Sadykov, T.M.: Computational problems of multivariate hypergeometric theory.Programming and Computer Software (2), 131–137 (2018)13. Sadykov, T.M.: The Hadamard product of hypergeometric series. Bulletin des Sci-ences Mathematiques (1), 31 (2002)14. Sadykov, T.M.: On the analytic complexity of hypergeometric functions. Proceed-ings of the Steklov Institute of Mathematics (1), 248–255 (2017)15. Stepanova, M.A.: Analytic complexity of differential algebraic functions. SbornikMathematics (12), 1774–1787 (2019)16. Stepanova, M.A.: On analytical complexity of antiderivatives. Journal of SiberianFederal University. Mathematics & Physics. (6), 694–698 (2019)17. Vitushkin, A.G.: On Hilbert’s thirteenth problem and related questions. RussianMath. Surveys,59