CC BY
5
DOI: 10.17516/1997-1397-2020-13-6-718-732 УДК 517.55; 517.9
Upper Bounds for the Analytic Complexity of Puiseux Polynomial Solutions to Bivariate Hypergeometric Systems
Vitaly A. Krasikov*
Plekhanov Russian University of Economics Moscow, Russian Federation
Received 10.06.2020, received in revised form 24.07.2020, accepted 20.09.2020 Abstract. The paper deals with the analytic complexity of solutions to bivariate holonomic hypergeometric systems of the Horn type. We obtain estimates on the analytic complexity of Puiseux polynomial solutions to the hypergeometric systems defined by zonotopes. We also propose algorithms 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.
Citation: V.A. Krasikov,Upper Bounds for the Analytic Complexity of Puiseux Polynomial Solutions to Bivariate Hypergeometric Systems , J. Sib. Fed. Univ. Math. Phys., 2020, 13(6), 718-732. DOI: 10.17516/1997-1397-2020-13-6-718-732.
1. Introduction and preliminaries
The notion of complexity is widely used in Mathematics and Computer Science in the context of several various abstract objects. The computational complexity of algorithms, the algebraic complexity of polynomials, the Rademacher complexity in the computational learning theory or the social complexity in the social systems are the concepts of great importance in the corresponding fields of science. The present work is devoted to the particular type of complexity -the analytic 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 initial formulation of Hilbert's 13th problem asks whether any continuous function of several variables can be represented as a finite superposition of bivariate functions [17]. The problem of finding similar representations for analytic functions has given rise to the theory of the analytic complexity. The main objects under consideration in this theory are the analytic complexity classes.
Definition 1 (See [2]). Let O(U(x0,y0)) denote the set of holomorphic functions in an open neighborhood U(x0,y0) of a point (x0,y0) G C2. The class Cl0 of analytic functions of analytic complexity zero is defined to comprise the functions that depend on at most one of the variables. A function f (x, y) is said to belong to the class Cln of functions with analytic complexity n > 0 if there exists a point (x0, y0) G C2 and a germ f(x, y) G O(U(x0, y0)) of this function holomorphic at (x0, y0) such that f(x, y) = c(a(x, y) + b(x, y)) for some germs of holomorphic functions a,b G Cln-i and c G Cl0. If there is no such representation for any finite n, then the function f is said to be of infinite analytic complexity.
* Krasikov.VA@rea.ru © Siberian Federal University. All rights reserved
Example 1. A generic element of the first complexity class Cl1 is a function of the form f3 (f1(x) + f2(y)). A function in Cl2 can be represented in the form /7 (f5(fi(x) + f2(y)) + fe(f3(x) + f4(y))), where fj(-) are univariate holomorphic functions, i = 1,..., 7.
For any class of analytic complexity Cln,n e N there exists a system of differential polynomials with constant coefficients An which annihilates a function if and only if it belongs to Cln.
Example 2 (See [2]). For a bivariate function f (x,y) consider the differential polynomial
A ( f) = f' ( f' )2 f''' — ( f' )2 f' f''' + f'' ( f' )2 f'' — f'' ( f' )2 f''
A1(J ) J x (f y ) J xxy (f x ) J y J xyy + J xy (f x ) J yy J xy (J y ) J xx.
This differential polynomial vanishes if and only if its argument f e Cl1.
The problem of defining whether a function belongs to an analytic complexity class is equivalent to computing the corresponding system of differential polynomials. Note that this is a problem of formidable computational complexity [4,11] and a direct approach to its solution appears to be inappropriate.
An important question is a possible connection between the classes of finite analytic complexity and hypergeometric functions. In this paper we consider hypergeometric functions as solutions of hypergeometric systems in the sense of Horn [8, 10]. We choose a matrix A e Zmxn = (Aj, i = 1,... ,m,j = 1,... ,n) and a vector of parameters c = (c1,..., cm) e Cm. We denote the rows of this matrix by Aj, i = 1,... ,m.
Definition 2. The hypergeometric system (or the Horn system) Horn(A,c) is the following system of partial differential equations:
xjPj(0)f(x) = Qj(0)f(x), j = 1,...,n, (1)
where
Aij — 1
Pj(s)= n n + cj + lf) ,
lAij 1-1
Qj(s)= n n (A*,*} + cj + ljl))
i:Aij <0 i(i)=0
d
and 9 = (01,..., 0n), 6a = x^——.
dxj
It has been conjectured in [14] that any hypergeometric function has finite analytic complexity. Hypergeometric systems of equations differ greatly from the differential criteria for the analytic complexity classes, but numerous computer experiments suggest that the conjecture is true in a lot of particular cases [6,7]. The case of hypergeometric systems with low holonomic rank has been considered 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 differentially transcendental functions, that is, functions that are not solutions to any nonzero differential polynomial with constant coefficients. Important examples of differentially algebraic functions of infinite analytic complexity have been presented in [15,16].
A bivariate hypergeometric system can be defined by an integer convex polygon and a complex vector of parameters as explained in the next definition.
Definition 3. Let li denote the generator of the sublattice {s G Zn : {Ai, s) = 0} and let hi be the number of elements in the set {A1,..., Am}, which coincide with Ai. Let us define the polygon P(A) (see [13]) as the integer convex polygon whose sides are translations of the vectors hili, the vectors A1,..., Am being the outer normals to its sides. We will say that the hypergeometric system Horn(A, c) is defined by the polygon P(A) and the vector c G C.
Definition 4. A polygon is called a zonotope if it can be represented as the Minkowski sum of segments.
In this article we investigate the analytic complexity of solutions to hypergeometric systems of equations (1) defined by zonotopes.
The present paper is organized as follows. In Section 2 we investigate particular cases of hy-pergeometric systems defined by zonotopes and analyze the analytic complexity of their solutions. We formulate and prove an estimate of the analytic complexity for Puiseux polynomial solutions to such systems in terms of the defining matrices and parameter vectors. In Section 3 we present algorithms for finding the supports of Puiseux polynomial solutions to hypergeometric systems and estimating the analytic complexity of polynomials. In Section 4 we consider examples of hypergeometric systems and estimate the analytic complexity of their solutions. Throughout the rest of the paper by «polynomial solutions to hypergeometric systems» we mean Puiseux polynomial solutions.
We use the Wolfram Mathematica package HyperGeometry for solving hypergeometric systems we investigate in this article. The package is available for free public use at https://www.researchgate.net/publication/318986894_HyperGeometry, the description of available functions is given in [12].
2. Hypergeometric systems defined by zonotopes
Let us consider the special case of hypergeometric systems defined by zonotopes. Numerous experiments suggest that the analytic complexity of polynomial solutions to such systems can be much lower than its estimate based on their supports (see [3, Proposition 4]).
The set of hypergeometric systems defined by zonotopes enjoys the following properties:
a) these systems are holonomic for generic values of parameters;
b) the holonomic rank of a hypergeometric system (see Theorem 2.5 in [5]) is given by
rank(Horn(A, c)) = d1d2 — ^^ vij, (2)
Ai,Aj lin. dependent
m
where dj = ^ Aij ,j = 1,2 and
i = 1 Aij > 0
( min(|Ai1Aj2|, \Aj1Ai2\), if Ai, Aj are in opposite open quadrants of Z2, ij 0, otherwise.
For the hypergeometric systems defined by zonotopes there is another formula for computing their holonomic rank (see Proposition 1 in [9]), which in some cases may be more suitable;
c) for any number of rows (ai, bi) belonging to the matrix A defining such a system, A contains the same number of rows (—ai, —bi). Thus the rows of A can be grouped into two matrices A, —A. This representation is in general not unique.
d) for a hypergeometric system defined by a zonotope one can always choose parameter values such that any solution to the resulting system is a polynomial (see [10, Proposition 6.5]). Namely, for such a hypergeometric system Horn(A, c), where the matrix A contains 2k rows, let a = (ai,... ,ak) be the part of the parameter vector c, corresponding to the matrix A (see the property (c) above),¡3 = (¡i,...,3k) be the part of this vector corresponding to -A. By Proposition 4.7 in [10] the general solution to Horn(A, c) is a polynomial if —ai — ¡i G N\{0} for i = 1,... ,k.
The simplest instance of a zonotope is a parallelogram. The analytic complexity estimate of the solutions to the systems defined by parallelograms is the basis for more complex cases.
Proposition 1. The analytic complexity of a solution to a hypergeometric system defined by a parallelogram cannot exceed 2.
Proof. The solutions to the hypergeometric system Horn(A, c) defined by a parallelogram have been described in Proposition 4.7 in [10]. For a bivariate system (n = 2) this formula leads to
(x-a11 x-a21 )ai (1 +
„ — aii x-a2i )—ai—Pi
(x-ai2 X-a22 )a2 (1 +;
-ai2 -a22
— a2-@2
where A
a11 a12 ),c = (aha2,pi,p2)- The monomials x—ai1 x—a21 and x—ai2x—a22 a2i a22 1
both belong to Cl1, thus for any univariate analytic functions $(•),$(■) the product
x^ ) ■ x—ai2x—a22 ) belongs to Cl2. □
The following example shows that the solutions to hypergeometric systems defined by more complex polygons can still have low analytic complexity.
Example 3. A .simple zonotope. Let us consider the hypergeometric system Horn(A', c') defined by the matrix A' = ^ 11 01 l) and the parameter vector c' =
(-23,22, -10,0, -9,0). Using the formula (2) we conclude that the holonomic rank of this system is equal to 3. The hypergeometric system Horn(A', c') is defined by the zonotope shown in Fig. 1.
+
+
Fig. 1. Polygon defining the system Horn(A', c'), and its representation as the Minkowski sum of segments
The support of the polynomial solutions to the system Horn(A', c') is shown in Fig. 2.
Let us consider the part of the solution p0(x, y) whose support is bounded by the straight lines parallel to the coordinate axes. Note that p0(x, y) contains 110 monomials (we do not put here the whole expression due to its large size) and the known estimates for polynomials [3, Proposition 4] imply that the analytic complexity of p0(x, y) does not exceed 5. Indeed, the support of p0(x, y) lies in the union of 10 lines parallel to the s axis. The analytic complexity of the polynomial
)
1
Fig. 2. The support for the solution of the system Horn(A', c')
whose support lies on a straight line parallel to an axis cannot exceed 1. Then the analytic complexity of the sum of k such polynomials cannot exceed 1 + \log2 k\, where by \x\,x G R we denote the smallest integer not exceeding x. Later we prove that in fact the analytic complexity of p0(x,y) does not exceed 3.
In general, appending a pair of rows (ai, bi), ( —ai, -bi) to the matrix defining a hypergeometric system is equivalent to adding a pair of parallel straight lines bounding the support of the solution in the exponent space. Let the hypergeometric system be defined by a parallelogram, and let
p0(x,y) = cs,t • xsyt be a polynomial solution of this system with the support S. Adding
(s,t)es
a pair of straight lines in the exponent space leads to the system whose solution is given by i \ r («is + ¡it + yi +1) s t ,,->., ^ s t
pi(x,y)= D r^s + 3it + Yi) ^ Cst ^ xSyt = £ («is + ^ + ^ xSyt =
(s,t)€S V 1 ' 1 '11 (s,t)€S
= (aiOx + ¡310y + Yi) Cs,txsyt = (aiOx + ¡3iOy + Yi)po(x,y).
(s,t)es
Using this formula repetitively we obtain the solution for k additional pairs of rows (ai, bi), (-ai, -bi):
pk (x,y) =
(n.
\j=i
(aj 0X + 3j 0y + Yj) po (x,y).
Thus the estimate for the analytic complexity of pk (x, y) depends on the analytic complexity
of p0(x, y). This dependence is described in detail in the following Proposition and its corollaries.
d d
Recall that we use the notation 9x = x — , = yjy and a.j , Yj G C, j = 1,..., h.
dy
s
Proposition 2. If f (x,y) G Cln then (aOx + /Oy + y)f (x,y) G Cl2n+1.
Proof. We use induction by n to show that (aOx + /Oy)f (x, y) G Cl2n. For n = 1 we can represent f (x,y) in the form f (x,y) = c(a(x) + b(y)).
(aOx + f39y)c(a(x) + b(y)) = c'(a(x) + b(y)) ■ (axa'(x) + /yb'(y)),
and this function belongs to Cl2 as a product of Cl\ functions. If the statement holds for all n < N, and f (x,y) belongs to ClN, which means it can be represented as f (x,y) = h(fi(x,y) + f2(x,y)), where fi(x,y),f2(x,y) G ClN-1, then
(aOx + /Oy )hf(x,y)+ f2(x,y)) =
= h' (fi(x, y) + f2(x, y)) ((aOx + /Oy )f(x, y) + (aOx + /Oy fx, y)).
Both of the functions f\(x,y) and f2(x,y) belong to ClNso the estimate of the analytic complexity for (aOx + /Oy)fi(x,y), i = 1, 2 is Cl2N-2. Then their sum belongs to Cl2Nand, after the multiplication of the result by h'(fi(x,y) + f2(x,y)) G ClN, the product belongs to Cl2N. Thus we conclude that for any n, if f (x,y) G Cln then (aOx + /Oy)f (x,y) G Cl2n. Adding Yf (x, y) G Cln to this expression we obtain a function in Cl2n+1. □
Corollary 1. For any f (x,y) G Cln the analytic complexity of
(jla Ox + /j Oy + Yj ^ f (x,y)
cannot exceed 2k (n + 1) — 1.
Corollary 2. Assume that the analytic complexity of a polynomial solution p0(x,y) to the hy-pergeometric system Horn(A,c) does not exceed n, S is a support of p0(x,y). Let the matrix A be obtained from A by appending k pairs of vectors (ai, bi), (—ai, —bi), vector c be obtained from c by appending 2k elements. Then the analytic complexity of a polynomial solution with the support S to the hypergeometric system Horn(A,c) does not exceed 2k (n + 1) — 1.
Example 3. (Continued). Let us use Corollary 2 to estimate the analytic complexity of a solution to the system Horn(A',c'). To do this, consider the system Horn(A', c'), defined by the matrix
T
A^^1! 0 1 ^ and the vector of parameters c' = ( — 10, —9,1,1). This system differs
from the original one only by the absence of the pair of straight lines with the normal vectors (1,1) and ( — 1, —1) bounding the support of the solution. Thus this support for the system Horn(A', c') coincides with the support of p0(x,y). Note that this system is defined by a parallelogram and hence by Proposition 2 the analytic complexity of its solutions cannot exceed 2. Computations show that the basis in the space of solutions to the system Horn(A', c') consists of the single function (x — 1)w(y — 1)9 G Cl\, and hence p0(x,y) G Cl3 by Corollary 2. The supports of two other solutions to Horn(A', c') lie on two parallel straight lines, so a linear combination of these solutions belongs to Cl3, and the general solution to Horn(A ', c') is a function in Cl4.
The following theorem is the main theoretical result of the paper. It contains the general estimate of the analytic complexity for polynomial solutions to hypergeometric systems defined by zonotopes.
Theorem 1. Let Horn(A,c) be a hypergeometric system defined by a zonotope. Assuming that the matrix A contains 2k rows, consider the matrices A and — A such that the union of their rows coincides with the set of rows of A. Let a be a part of the parameter vector c corresponding to the matrix A, ¡3 be a part of this vector corresponding to —A, and define the vector c = (C\,..., Ck) by Ci = —a — ¡i.
If ci £ N\{0}, i = 1,... ,k, then the analytic complexity of the general solution to Horn(A, c) does not exceed
min[3 • 2k-2 - 1 +
log2 k(k0 , 2 + \log2C max, C + 1)1 + \ ^(k - 1)1) ■
2 i=1,...,k I
Proof. For any system defined by a parallelogram the condition ci G N\{0} provides the existence
of a polynomial solution (see [10, Proposition 4.7] and the proof of Proposition 2.). Appending
of the rows (ai, bi), (-ai, -bi) to the matrix defining the hypergeometric system affects only the
coefficients of this solution but not its support. Without loss of generality we can choose a vector
of parameters c such that the support of the general solution to Horn(A, c) coincides with a union
of supports of the solutions to a finite number of systems defined by parallelograms (see proof of
Proposition 6.5 in [10]). Thus the condition ci G N\{0} provides the existence of a polynomial
basis in the space of solutions to Horn(A, c)■
The matrix A contains 2k rows, so supports of the solutions are bounded by k pairs of straight
lines. Let us assign a natural number from 1 to k to each pair of lines. The union of these supports k(k 1)
is a subset of-^- parallelogram intersections (it is the sum of an arithmetic progression),
each intersection we denote as □ijj, where i G {1, ■ ■ ■, k} and j G {1, ■ ■ ■, k} are numbers assigned to pairs of straight lines which form the intersection, i < j. For any (i,j) G {1,■■■,k}'2 the solution to Horn(A, c) whose support lies in the intersection □ijj belongs to Cl3.2k-2-1 (by
k(k 1)
Corollary 2). The analytic complexity of the sum of-2-functions in Cl3.2k-2-1 (that is, the
analytic complexity of the general solution to Horn (A, c)) cannot exceed estimA |J =
\i,j = 1 ' J
k(k — 1) 1 log2-2- (see [3, Section 5]).
On the other hand, there is the estimate based on the number of parallel straight lines containing the points of the support (see Proposition 4 in [3]). While the analytic complexity of any polynomial with the support belonging to a straight line does not exceed 2, the number of these
lines corresponding to the i-th row of A equals ci +1 ■ Thus for any i the analytic complexity of the
it
part of the solution whose support belongs to |J □i-j cannot exceed 2+ |"log2( max ci + 1)].
j=1 ' i=1,...,k
Note that there is no need to use all of k pairs of bounding straight lines to estimate the analytic complexity of the general solution this way, since k - 1 pairs already bound the whole
support of the solution. The sum of k — 1 elements in Cl2+^og2( max cannot exceed
i=1 ,...,k
estim2[ IJ dij 1 = 2 + |"log2 ( max ci + 1 )] + \log2(k — 1)] ■ The minimal of the numbers
\i j=1 > \i=1,...,k /
estim1 □i j , estim2 □i j is the sought estimate. □
An example of using the estimate given in Theorem 1 is shown in Fig. 3. Note that there are three sets of parallel lines, each corresponding to one of the ci ■ For each of the parallelogram intersections there are 2 estimates: estim1(di,j), based on Corollary 2 and estim2(dijj) based on Proposition 4 in [3].
3 • 2k-2 - 1 +
estimi (□! 3) = 3 • 2k 2 — 1
eitim2(Dij3) = 2 + flog2 (min(ci + 1,63 + 1))! = 3
estim-i (□i 2) = 3 • 2k 2 — 1
estim2(^i,2) = 2 + flog2(min(ci + 1, ¿2 + 1))! = 3
estimi(^2 3) = 3 • 2k 2 — 1
estim2(^2,3) = 2 + riog2(min(c"2 + 1, 63 + 1))! = 4
Fig. 3. The analytic complexity estimate for a polynomial solution to a simple hypergeometric system
We order ci by the ascension and choose v to be a vector with the elements vi = = min (2 + \log2(ci + 1)1, 3 ■ 2k-2 — 1+ \log2(k — i)"|) ,i = 1,... ,k — 1. To find more accurate value for the analytic complexity estimate from Theorem 1, one could use Algorithm 2 from Section 3 using v as an input vector. The general estimate from Theorem 1 can be rough, if values of ci vary greatly for different i. For example in Fig. 3 estim2(^i,2) = estim2(^i,3) = 3, estim2(^2,3) = 4, and for the general estimate we use the maximal of these values. The vector v in this case provides the choice of the better estimate.
3. Algorithms of analytic complexity estimation
The following algorithm allows one to compute the analytic complexity of any given bivariate polynomial.
Algorithm 1: Finding an analytic complexity estimate for a polynomial
Input: p(x, y) - a polynomial, x,y G C.
Output: N - an estimate for the analytic complexity of p(x,y).
1 result ^ 0
2 short ^ {}
3 polys ^ {pi(x,y)\p(x,y) = J2pi(x,y), Supp pi(x,y)||Supp pj(x,y)Vi,j}
8 short = short U curr
9 N ^ 2+ \Log2(result)1
The main advantage of this algorithm compared to the existing ones is its ability to distinct the powers of lower degree polynomials included in the original polynomial as summands. Without this feature, even the analytic complexity of the function like p(a(x) + b(y)) G Cl\, where p(t), a(x),b(y) are univariate polynomials, is estimated based on its support, which becomes very
4 for p G polys do
5 curr = getShort(p)
6 if curr ^ short then
7 result += 1
inaccurate with the growth of degree of p(t).
The input of the function getShort() is a homogeneous polynomial and the output contains elements of its decomposition into the sum of powers. Note that the definition of polys assumes the ambiguity of the representation of the polynomial as the sum of finitely many polynomials supported in parallel straight lines. Any of such representations yields an estimate, but some of them may be better than the other ones.
To estimate the analytic complexity of the general solution to the hypergeometric system from Theorem 1 one can use the following algorithm.
Algorithm 2: Finding an analytic complexity estimate for a sum of functions
Input: c = {c1, c2,..., cn} - a set of known estimates of the analytic complexity values for bivariate functions f1 (x, y), f2(x,y),..., fn(x, y), where (x, y) £ C2.
n
Output: N - an estimate for the analytic complexity of the function ^ fi(x, y).
i=i
1 while c contains more than 1 element do
2 find 2 minimal elements of c, namely, ci and cj.
3 c = (c U {max(ci, cj) + l})\{ci, cj}.
4 N ^ only element of c.
Algorithm 2 is finite, since at each step the number of elements in c decreases by l.
The following algorithm allows one to find the support of a polynomial solution to a given hypergeometric system defined by a zonotope, provided that such a solution exists. The algorithm is based on Proposition 4.7 in [10].
Algorithm 3: Constructing the support of a polynomial solution to a hypergeometric
system
Input: the matrix A, the parameter vector c for the hypergeometric system Horn(A, c) defined by a zonotope
Output: supp - the support for the polynomial solution to Horn(A, c).
1 supp ^ {}
2 find A : rows(A)U rows—A) = rows(A)
3 for (ri, rj) C rows (A), i < j do
4 Ai,j ^ (ri,rj)T
5 a ^ elements of c corresponding to (ri,rj)
6 3 ^ elements of c corresponding to (—ri, —rj)
7 if —a.j — 3j > 0 for j = l, 2 then
( -A-1 ( -A-1 \-a1-l31 ( _A-1 \ ft 2
8 supp = supp U Supp I x a 11 + x eM (l + x e2 j
9 else
10 the general solution to Horn(A, c) is not a polynomial
For some pairs of rows ri,rj- the solution to the corresponding system defined by a parallelogram is not a polynomial. In this case, a part of the basis in the solution space can still consist of polynomials, and their supports can be found by means of Algorithm 3.
4. Examples
Example 3 (Continued). Let us replace the parameter vector c' in the system Horn(A',c') by
the vector (k, 0,0,0,0,0). The corresponding system is given by
x9x(9x + dy + k) - 9X(9X + dy),
yOy(Ox + 0y + k) - dy (9x + dy).
x k-1 (-1)j y k-1 (-1)j
A basis in its solution space is given by 1, log--+ J2 -j, log--+ J2 -j,
x - 1 j=i J(x - 1)j y - 1 j=i J(y - 1)j so there is no polynomial basis for these parameter values. Nevertheless, the analytic complexity of the general solution is equal to 1.
The present example shows that the analytic complexity of solutions to hypergeometric systems can be heavily dependent on parameter vectors defining these systems. A resonant choice of their parameters can drastically reduce the analytic complexity of general solutions to such systems.
Example 4. An octagon zonotope. Consider Example 6.8 in [10]. In order to find the analytic complexity of a polynomial solution to the hypergeometric system defined by the matrix
. _ ( 1 -1 -1 1 -3 3 2 -2 )T
A V 2 -2 1 -1 -2 2 -1 1 )
and the vector of parameters c _ (3, -5, -2,1, -2, -1, -1, -1) we can use the basis of the solutions to this system, computed in [10]. There are 3 solutions whose analytic complexity is equal to 2, and 28 solutions in Cl\, two of them also belonging to Cl0. Therefore the analytic complexity of the general solution to this system cannot exceed 7. Note that this estimate is based on a trivial grouping of the basis functions into pairs, but the very specific structure of the solution support makes it possible to show that the analytic complexity does not exceed 6.
Let us estimate the analytic complexity of the general solution to this system using Theorem 1. The vector c, ordered by the ascension, is (1, 2,2,3). Then the vector v _ (3,4,4) (it includes only support-based estimates, because of low values of the elements of c), and, by using Algorithm 2, we conclude that the general solution belongs to Cl6.
Futhermore, we can estimate the analytic complexity of a solution to any hypergeometric system we obtain by appending a pair of rows to A (the only condition is that these rows are not collinear to the rows of A). Note that this estimate does not depend much on the difference between new parameters. If this difference is big, it becomes the last element of the ordered vector cc, and does not affect the new vector v, the new element of the vector v is equal to 2 + [log2(3 + 1)] _ 4, and the resulting analytic complexity is 6. On the contrary, if this difference is low, for example, if it is equal to 1, the new vector c _ (1,1,2, 2, 3), the new vector v _ (3,3,4,4), and the analytic complexity is also equal to 6. Thus we conclude that appending 2 rows to the matrix A does not affect the analytic complexity of the solution to the system.
Example 5. A decagon zonotope. Consider the hypergeometric system Horn(Ai, c1), defined by the matrix
(-1 1 0 0 -2 2 3 -3 3 T (3)
^00 -111 -11 -12 -2 ) (3)
and the parameter vector c\ _ (-1,0,4, -5,1, -4, -9,6, -4,0). The zonotope defining the matrix 3 is shown in Fig. 4.
By Theorem 2.5 in [5] the holonomic rank of the system Horn(Ai, ci) equals 34. The support to the solution to this system computed by the means of Algorithm 3 is shown in Fig. 5.
A polynomial basis in the solution space to Horn(A\,c\) consists of the 4 monomials x^ xi^/3 x3 x8/3
—9, —^ ^, —^ and 30 polynomials y9 y8 y3 y2
Fig. 4. The zonotope which defines the matrix (3)
1 5643 247095 329460 27455 82365 741285 724812
C H~ A r^r^r^-1 Q r-v ^ A .A Q A ^ O
xy6 637xy5 8281xy4 8281xy3 286y4 49y3
49y2
7y
20y3 4y2 4y2 18y 1 63x 35x 5 5
11y
i2
33y
ii
297y
10
380x 24y9
3969y5/2 1323yr/2 51 3/2
--1----y3/ 2 + \/y,
41990x 16796x 55 vy'
115311x 38437x
5
99x + 3x + 81 +y ,
100555x 5915x 1550775x7/2y5
+
3y
9
„8
- + yl + 3H+ y6
1105 26 143 y '
82808479
44
1547x4y5 91x4y "l03455
87y5
91x3y5 + 3 4 +xy,
+ 36575y4 +
+
6840 5220y5
31465x'9/2y5 61400001
806y5
+ x5/2y4
84656y4 +
y
4
129x8/3 735x5/3
y4 44y5 33y5
82x7/3 275561x10/3 2392x4/3 x7/3
+
1183x3 182x2
4 2 5
xy — 13 xy ,
5175x7/2y4
x8/3' +
^ 2 ' x2
89947
13/3
x
y6 16/3
+
y
,8
+
451x13/3 261y5 ,
1378x16/3 451y7 ,
21 2/3 5 2/3 4 119 5/3 4
--x 1 y + x ' y +--x 1 y ,
46 y y 286 y '
11985 8/7 2/7 14382x8/7 -x8/7y2/7 +
299
114774x6/7y5/7 28405
253y5/7
+
x
8/7
1188x6/7 + _ ^ +
y12/7 x6/7
65y2/7
y
9/7
y
c4/7 32680x11/7 1558
--I---I--'
6/7 8613y6/7 261 "
6/7
4/7
12 4/3 5 , 4/3 4 364 tyxy
--x 1 y + x ' y--
247 y y 1045
1200x2/7 345x<9/7 x9/7 + _ +
1643y3/7 31y3/7 y10/71
x10/7 731x3/7 1763x10/7
+ ^ ^ +
u8/7
638
^, 169 x5/7y3/7 + ^ + 4/7
5/7
754
65x12/7
2x 7y
x10/3 261x10/3 y4 + 238y3 ,
4/7 136y4/7
1 c 2 3, 5 2 2 45 2 , 2
— 66 +7x y — 28x y + x
9/5 3/5 1287x9/5y8/5 55913x9/5y13/5
x y TT^TTa +
150
x1i/5y2/5 4301x11/5y7/5 + 232254x11/5y12/5
4277 68
1056419
x8/5y6/5 +
1634
5824x13/5y6/5
432837
346408
1064x8/5y11/5 2829 :
x12/5y4/5 _ 68x7/5y9/5 _ x12/5y9/5
x y 19x y 231x y ,
8x
5
21x
x14/3 828x14/3
85y6
585488x11/3 21758x14/3
+
y7 + 15y6 55y6 2488324x11/3
4
4
91x
182x
15y5 24y4
4
48825y5
23715y5
35805y4
There are 14 functions in Cl1 and 20 functions in Cl2\Cl1 among these polynomials.
The analytic complexity estimate of the general solution to Horn( A1 ,c1) obtained by grouping these functions into pairs is Cl7. Theorem 1 gives a better estimate: since c = (2, 2, 3,3,4),
y
v _ (4,4,4,4), it follows that the general solution belongs to Cl6.
The following examples present hypergeometric systems defined by polygons other than zono-topes whose solutions have low analytic complexity.
Example 6. A pentagon. The matrix ^ ° 00 0 1 0 ) and the vector of parameters ( 4, 0, 0, 1, 2, 1, 2) define the hypergeometric system
x(Ox + Oy -4)(Ox -1) -Ox(Ox -2),
y(Ox + Oy - 4)(Oy - 1) - Oy (Oy - 22). ()
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
x2y2, 1 - 4x - 4y +12xy, 6x2 - 4x3 + x4 - 12x2y + 4x3y, 6y2 - 12xy2 - 4y3 + 4xy3 + y4.
The first and the second of these polynomials belong to Cl\, the third and the fourth belong to Cl2. Thus the general solution is a function in Cl4.
Example 7. A trapezoid, high holonomic rank. A basis in the solution space of the hypergeo-metric system with holonomic rank k defined by the operators
xOk-1(Ox + Oy) - (-1)kOk, y(Ox + Oy) + Oy.
is given by {logj ((y + 1)/x), j _ 0,... ,k - 1} (see Fig. 6). The generating solution equals logfc-1 ((y + 1)/x). Thus the general solution to this system belongs to Cl\ by the conservation principle. This example shows that the analytic complexity of solutions to hypergeometric systems with high holonomic rank can still be low.
ià.
b)
b) polygon defining the system (4)
Example 8. A triangle with no symmetries. The hypergeometric system х(вх + в y - 4)(вх + 2ву - 4) - (2вх + 3ву - 4)(2вх + 3ву - 5),
(5)
у(вх + в у - 4)(вх + 2ву - 4)(вх + 2ву - 3) - (2вх + 3ву - 4)(2вх + 3ву - 5)(2вх + 3ву - 6)
is holonomic and its holonomic rank equals 6. The pure basis in its solution space is given by the Laurent polynomials
x-4y4, х-2 y3, x7y-3, x8y-4, 3y2 +2x-1y2,
6x2 + I2x3 + x4 + 4x5y-2 + 6x6y-2 - I2x4y-1 - 4x5y-1 - I2xy - 4x2y.
In the Fig. 7 the small filled circles correspond to monomial solutions, the two empty circles indicate the binomial solution and the big filled circles correspond to the remaining polynomial solution. The analytic complexity of the general solution to the system (7) does not exceed 5.
This research was performed in the framework of the state task in the field of scientific activity of the Ministry of Science and Higher Education of the Russian Federation, project "Development of the methodology and a software platform for the construction of digital twins, intellectual analysis and forecast of complex economic systems", Grant no. FSSW-2020-0008.
Fig. 6. a) the supports of solutions to the system (4);
References
[1] V.I.Arnold, On the representation of continuous functions of three variables by superpositions of continuous functions of two variables, Sbornik Mathematics, 48(1959), no. 1, 3-74.
[2] V.K.Beloshapka, Analytic complexity of functions of two variables, Russian J. Math. Phys., 14(2007), no. 3, 243-249.
[3] V.K.Beloshapka, Analytical complexity: Development of the topic, Russian J. Math. Phys., 19(2012), no.4, 428-439.
[4] V.K.Beloshapka, On the complexity of differential algebraic definition for classes of analytic complexity, Math. Notes, 105(2019), no. 3, 323-331.
[5] A.Dickenstein, L.F.Matusevich, T.M.Sadykov, Bivariate hypergeometric D-Modules, Advances in Mathematics, 196(2005), 78-123.
[6] A.Dickenstein, T.M.Sadykov, Algebraicity of solutions to the Mellin system and its mon-odromy, Dokl. Math., 75(2007), no. 1, 80-82. DOI: 10.1134/S106456240701022X
[7] A.Dickenstein, T.M.Sadykov, Bases in the solution space of the Mellin system, Sbornik Mathematics, 198(2007), no. 9, 1277-1298.
[8] J.Horn, Uber die Konvergenz der hypergeometrischen Reihen zweier und dreier Veranderlichen, Math. Ann., 34(1889), 544-600.
[9] V.A.Krasikov, Analytic complexity of hypergeometric functions satisfying systems with holo-nomic rank two, Lecture Notes in Computer Science, 11661(2019), 330-342.
10] T.M.Sadykov, S.Tanabe, Maximally reducible monodromy of bivariate hypergeometric systems, Izv. Math., 80(2016), no. 1, 221-262.
11] T.M.Sadykov, Beyond the first class of analytic complexity, Lecture Notes in Computer Science, 11077(2018), 335-344.
12] T.M.Sadykov, Computational problems of multivariate hypergeometric theory, Programming and Computer Software, 44(2018), no. 2, 131-137. DOI: 10.1134/S0361768818020093
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Верхние границы аналитической сложности решений двумерных гипергеометрических систем в классе многочленов Пюизо
Виталий А. Красиков
Российский экономический университет им. Г. В. Плеханова
Москва, Российская Федерация
Аннотация. В статье исследуется аналитическая сложность решений двумерных голономных гипергеометрических систем типа Горна. Получены оценки аналитической сложности решений в классе многочленов Пюизо для гипергеометрических систем, заданных зонотопами. Также предложены алгоритмы для оценки аналитической сложности многочленов.
Ключевые слова: гипергеометрические системы дифференциальных уравнений в частных производных, голономный ранг, полиномиальные решения, зонотопы, аналитическая сложность, дифференциальный многочлен.
