The Newton polytope of the optimal differential operator for an algebraic curve Текст научной статьи по специальности «Математика»

УДК 510.52+517.554+517.953

The Newton Polytope of the Optimal Differential Operator for an Algebraic Curve

Vitaly A. Krasikov*

Institute of Mathematics and Computer Science, Siberian Federal University, Svobodny, 79, Krasnoyarsk, 660041

Russia

Timur M. Sadykov^

Department of Information Technologies Russian State University of Trade and Economics,

Moscow, 125993,

Russia

Received 30.12.2012, received in revised form 10.01.2013, accepted 25.02.2013 We investigate the linear differential operator with polynomial coefficients whose space of holomorphic solutions is spanned by all the branches of a function defined by a generic algebraic curve. The main result is a description of the coefficients of this operator in terms of their Newton polytopes.

Keywords: algebraic function, minimal differential operator, Newton polytope.

Introduction

To find relations satisfied by a given special function is a difficult and important problem in the theory of special functions of mathematical physics. The relations in question can involve derivatives, integrals, finite differences etc. Knowing a global relation for a special function that is defined locally (e.g. by means of a series converging in a neighbourhood of a point) allows one to deduce global properties of that function. From this point of view, linear differential equations with polynomial coefficients are of particular interest. One of the reasons for this is the difficult problem of computing the analytic continuation along a given path of a locally defined special function. By identifying such a function with a solution to a system of linear differential equations with polynomial coefficients which does not have any "extra solutions" one can use standard techniques for investigating the analytic continuation of the function under study (see, for instance, [14]). Here by "extra solutions" we mean the solutions which are not branches of the function under study, that is, which cannot be obtained from it by means of analytic continuation. Observe that every germ of a (multivalued) analytic function satisfies a relation with entire (in particular, polynomial) coefficients provided that this relation is valid for one of its germs in a neighbourhood of some fixed nonsingular point.

The culmination of this approach is the Wilf-Zeilberger algorithmic proof theory (see [18] and [19]) based on holonomic systems of equations. In the present paper we thoroughly investigate the special case when the function under study is algebraic and the holonomic system consists of a single ordinary linear homogeneous differential equation with polynomial coefficients. Despite all simplicity, this setup leads to formidable computational challenges.

The 21st problem in the Hilbert list was solved in 1989 by A.A. Bolibrukh who proved that it is in general not possible to construct a linear fuchsian system of differential equations with

* vkrasikov@sfu-kras.ru tSadykovTM@rsute.ru © Siberian Federal University. All rights reserved

a prescribed monodromy group (see [1]). However, the problem of effective computation of a system of differential equations (and, in particular, of a single differential equation) with a prescribed branching of solutions (whenever this is possible) remains open and is in the focus of intensive research, see [4,6,9]. The computer algebra system Magma has a built-in command DifferentialOperator for finding such operators (see [5]). Another powerful tool for finding linear differential equations (both homogeneous and inhomogeneous) for algebraic functions is the gfun package developed by B. Salvy et. al. (see [4]).

In the present paper we describe an algorithm which allows one to compute annihilating operators for an essentially larger class of algebraic functions (see Examples 4 (4), 9, 10 and 11). We also provide a combinatorial characterization of the coefficients of the optimal annihilating operators in terms of their Newton polytopes.

It is well-known that an ordinary linear differential equation with a prescribed solution space can be found by means of the wronskian of a basis of this space. However, from the computational point of view, the wronskian-based representation of the differential equation for an analytic function (which is, in general, defined only locally) is merely a nonconstructive existence theorem. There are three main reasons for this. First, to form the wronskian, one needs to choose a basis in the space of germs of the given function at a nonsingular point. This requires computing the analytic continuation of the given function along any path, which is, in general, a difficult problem. Secondly, to evaluate a determinant containing high-order derivatives of a given special function is a task of a great computational complexity. Finally, extracting the polynomial coefficients of the desired differential operator out of the obtained combination of algebraic functions requires the full use of modern methods of computer algebra. For instance, to compute the differential operator for the roots of the generic monic cubic by means of the wronskian is already a challenge (see example in Sec. 5 in [11]). In the general case, the wronskian-based construction is not suitable for computation since no effective means of simplifying expressions which contain high-order derivatives of special functions are presently known.

The present paper provides an algorithm for constructing the optimal (that is, of the smallest possible order) linear homogeneous differential equation with polynomial coefficients for a univariate algebraic function y = y(x) implicitly defined by the equation

ym + ai ymi + ... + an ymn + x = 0. (1)

The proposed method is a development of the ideas of the work [11]. It allows one to reduce the problem of computing the annihilating operator for an algebraic function to the problem of finding a basis in the syzygy module of an ideal in the ring of multivariate polynomials. The presented algorithm differs from other methods (both recent and classical, see [4,6,9]) in its primary field of application (it deals with generic algebraic equations), in the underlying concept (holonomic systems of partial differential equations and noncommutative elimination) and the complexity of differential operators that it can efficiently produce. The capabilities and limitations of the proposed algorithm are summarized in Table 1.

The authors are thankful to D.Zeilberger for helpful explanations giving insight into holo-nomic systems approach, to M. Singer for comments on Galois theory and to L. Matusevich for fruitful discussions.

1. Annihilating operators for solutions to holonomic systems of differential equations

In what follows we will denote by Dn the Weyl algebra of differential operators with polynomial coefficients in n variables x = (xi,..., xn) e C . This algebra is generated by the operators xi,..., xn, dxi,..., dXn satisfying the relations dXi o xj — xj o dXi = Sj. Here "o" denotes the

composition of differential operators. The Weyl algebra is simple (see Chapter 1 in [2]). When speaking about ideals in the Weyl algebra, we will always mean its left ideals.

The following basic statement is well-known but not easy to find in the literature in the following explicit form. It can be deduced from Theorem 2 in [17]. It also follows from Theorem 1.4.12, Proposition 1.4.9, and Lemma 2.2.3 in [15].

Proposition 1. For any holonomic left ideal I C Dn and any i £ {1,...,n} there exists a nonzero operator Pi £ I, all of whose derivatives are with respect to the variable xi, that is, an operator of the form

Ni

Pi ^ ^ aij (x1 ? • • • ? xn ) dXi • j=1

The following statement is a consequence of the results in [10] and [12]. It shows that algebraic functions defined by generic algebraic curves are annihilated by holonomic ideals in Dn.

Theorem 2. Any germ of the algebraic function y(x0, xi, ... ,xn) implicitly defined by the relation

xnyn + xn-iyn-i + ... + xiy + xo = 0, (2)

satisfies the holonomic system of differential equations

dxidx, y = dXk 3Xl y, whenever i + j = k + l, n n (3)

i xi dxiy = -y and J2 xi dxiy = 0. ( )

i=o i=o

Conversely, any holomorphic solution of (3) defined locally in a neighbourhood of a nonsingular point is a linear combination of germs of the function y(x0, xi, ..., xn) at this point.

The system of differential equations (3) is a special instance of the Gelfand-Kapranov-Zelevinsky hypergeometric system introduced in [10]. Its "dehomogenized" version for an algebraic curve with affine parameters was investigated by Mellin in [13].

Recall that the Nilsson class comprises (multi-valued) analytic functions of several complex variables which have finite determination and moderate growth in arbitrary neighborhood of any of their singularities (see 4.1.12 in [3]). Here by the determination of a multi-valued analytic function we mean the number of its linearly independent germs in a neighborhood of a generic point in its domain of definition. The determination of an analytic function of one complex variable which lies in the Nilsson class and has finitely many singularities in C coincides with the smallest possible order of an ordinary linear homogeneous differential equation with polynomial coefficients satisfied by this function.

Theorem 2 together with Proposition 1 imply the existence of a linear differential operator with polynomial coefficients whose space of local holomorphic solutions at a nonsingular point is spanned by the roots of the generic algebraic equation (2) and all of whose derivatives are with respect to x0. This operator is defined uniquely up to a sign. We will say that this operator is optimal for the given generic algebraic curve.

Example 3. Consider the algebraic function y(x0, x1; x2) defined as the solution to the quadratic equation x2y2 + x1y + x0 = 0. By Theorem 2, any of its branches lies in the kernel of any operator in the ideal J with the generators

A = dx00x2 - , B = xidxj + 2x2dx2 + 1, C = x0dx0 + xid^ + x2dx2.

Since the determination of the function y(x0,xi,x2) equals 2, Proposition 1 yields the existence of a second order differential operator P £ J, all of whose derivatives are with respect to x0.

Using the notation = x4dXi, we can write the expansion of this operator with respect to the basis of J in the form

P = xoxix2A — ((xi — 2x0x2)^0 + xox2#i )B + ((xi — 4x0x2)^0 + 2xox2^i)C = = x0 ((xi — 4x0x2)dX0 — 2x2dxJ .

Of course, this optimal differential operator is only a monomial multiple of the wronskian of the roots of the initial algebraic equation.

In the next section we describe the algorithm for computing the optimal annihilating operator for an arbitrary algebraic function satisfying an equation of the form (1). This will, in particular, perform the noncommutative elimination of all the derivatives except for dXi in the holonomic ideal (3) by means of methods of commutative algebra only.

2. Computing the annihilating operator for a given algebraic function

We begin by computing the determinations of some elementary functions and the corresponding differential equations.

Example 4. (1) Any rational function f = p(x)/q(x), where p(x), q(x) e C[x], has determination 1 and satisfies the first-order differential equation pqf' = (p'q — pq')f.

(2) The function f = x" also has determination 1 for any a e C since its analytic continuation e2niaf around the only finite singularity x = 0 is proportional to f. It satisfies the differential equation xf' = af.

(3) The function f = ln x has determination 2, since its analytic continuation along any path can be written in the form ln x + 2nki, k e Z. Thus any germ of f at a nonsingular point lies in the two-dimensional linear space with the basis {1, ln x}. The second-order differential equation with polynomial coefficients satisfied by f has the form xf'' + f' = 0.

(4) The algebraic function y = y(x) implicitly defined by the relation y5 + ay + x = 0 has determination 4 (see Theorem 5 below) and satisfies the differential equation (256a5 + 3125x4)y(4) + 31250x3y(3) + 73125x2y(2) + 31875xy' — 1155y = 0.

(5) Finally, the function 1/ lnx has infinite determination since its germs {1/(lnx + 2nki)}fcez are linearly independent. This implies, in particular, that this function does not satisfy any linear homogeneous differential equation with polynomial coefficients.

In the present section we describe an algorithm for computing the optimal annihilating operator for the roots of a generic algebraic equation with symbolic coefficients, that is, an equation of the form (1). The roots of the equation a0ym + aiymi + a2ym2 + ... + anymn + an+i = 0 (regarded as functions of a = (a0,...,an+i)) satisfy the holonomic A-hypergeometric system with the vector of parameters (0, —1) (see [16]), where

A := f1 1 ... 1 0'

m mi . . . mn 0

Namely, it is the left ideal in the Weyl algebra C[a0,..., an+i, d0,..., dn+i] generated by the

toric operators 3U — dv for u, v e N + with A • u = A • v, and the Euler operators ^^ ajcj and

n+1

j = 0

ma0 + ^^ mj a^ dj + 1. Thus by Proposition 1, there always exists a linear differential operator

j=i

n

with polynomial coefficients in ao,..., an+i, and all of whose derivatives are with respect to an+1. Setting a0 = 1 and an+1 = x we obtain the annihilating operator for the solutions of (1). Using noncommutative elimination theory, one can compute this operator in a way similar to that in Example 3. In the special case of a trinomial equation (that is, for n = 1) the desired operator is a right factor of the Mellin differential operator found in [13].

The following theorem gives the order of the annihilating operator.

Theorem 5 (E. Cattani, C.D'Andrea, A. Dickenstein [7]). The number of linearly independent (over the field of complex numbers) germs of the solutions to the equation (1) at a generic point x £ C and for generic values of the parameters (a1,..., an) £ C is given by

m — 1 +

R(m, mi,. .., mn)

mi m—1

m

GCD(m, mi,..., mn)' Here [] denotes the integer part of a real number.

if GCD(m, mi,..., mn) = 1, if GCD(m, mi,..., mn) > 1.

The following theorem is the foundation of our algorithm for computing optimal annihilating operators.

Theorem 6. Let Sj = sj(x, a1,..., an), i = 1,..., m be the roots of the algebraic equation (1)

m

and denote P(t) = (t — Sj). For every k = 1,..., m we define the ideal in the polynomial

i=1

ring with m + n +1 variables C[s1, ..., sm, a1, ..., an, x] to be

( —1)l(l — 1)!(n(sfc — si)

2m-1

1

P (t)1

I = 1, .. ., R(m, mi, .. ., mn) I .

The vector of polynomial coefficients of the optimal annihilating operator for the algebraic function defined by (1) lies in the following syzygy module of the quotient of the ideal with respect to the Vieta relations:

Syz(lfc/( Sm-mj (si,...,sm) — ( — 1)m mj Oj , j = 1,...,n;

Sm-k (si,...,sm), for k ^ {0, mi,m2 ,...,m„}; Sm(si, . . . , Sm) — ( — 1)mx)) .

(4)

Here Sj(s1 ,...,sm) is the elementary symmetric polynomial of order j in the variables (s1,..., sm). In the sequel we will denote the ideal generated by the Vieta relations by V.

Proof. Let x £ C be a point outside of the zero locus of the discriminant of the left-hand side in (1). Let (a1,..., an) £ C be a generic vector of parameters and let (x, a1,..., an) denote the k-th branch of the solution y(x, a1,..., an) to the algebraic equation (1). Let us now denote by D the differential operator D = dsi ... dSm. Using the well-known contour integral representation for a solution to a univariate algebraic equation (see Section 5 in [11]) we conclude that the generators of the ideal are polynomial multiples of the derivatives of the solution to (1):

dX yk (x, ai

( —1)1 ((I — 1)!)m-i

D

,i-1

a„) = ( —1)l(l — 1)! tres -L.

i=Sfc P(t)1

1

(sk — si) ... [k] ... (sk — Sn)

(—1)1

((I — 1)!)'

-i

D

,i-i

1

res

= (—1R

^l-i

i=sfc P(t)

1

Sk ((Sk — Si) ... [k] ... (Sk — Sm))1

t=s

k

^ (I + ii - 1)!... [k]... (I + - 1)!

n + [fc] +in=l-i ((I — 1)!)m-2 ii!... [k]... im! (sfc — si)l+ii ... [k]... (Sfc — sn)l+- '

This shows that the generators of the ideal are indeed elements of the ring ai, ..., an, x]. By Theorem 5 the determination of the solution to (1) equals R(m, mi, ..., mn). Thus by Theorem 2 and Proposition 1 there exists a linear differential operator with polynomial coefficients (in x, ai,..., an) all of whose derivatives are with respect to x and whose space of holomorphic solutions at a generic point is spanned by the branches of y(x, ai,..., an). For the sake of computational efficiency we factor out the Vieta relations. This increases the number of variables involved in the generators of the ideal but decreases their degrees. The desired differential operator is a relation between the derivatives of y(x, ai,..., an) with polynomial (and thus single-valued) coefficients. By the conservation principle for analytic functions the same relation must be satisfied by any of the germs of y(x, ai,..., an) at a nonsingular point. Thus the coefficients of this relation lie in the syzygy module (4). □

Observe that the elements of the syzygy module (4) are polynomial vectors whose entries in general depend on all of the variables

s i, ..., s ^^, a i,. .., an, x*. The proof of Theorem 6 implies that there exists an element of (4) whose entries only depend on ai,..., an, x. It can be found by means of the following algorithm.

Algorithm 7. The actual computation of annihilating operators for algebraic functions was organized as follows:

1. Compute the basis of the ideal Ii defined in Theorem 6.

2. Using the lexicographic order of the variables si,..., sm compute the Grobner basis of the ideal V defined by the Vieta relations (as defined in Theorem 6).

3. Perform polynomial reduction of the generators of the ideal Ii by means of the Grobner basis of the ideal V. That is, at this step, we use the Vieta relations as much as possible in order to simplify the generators of V.

4. Factorize the obtained family of polynomials. The result has a very specific structure: it is a family of polynomials in C[si, ..., sm, ai, ..., an,x] whose elements are symmetric with respect to s2,..., sm. Using the Grobner basis of the ideal V, reduce them to polynomials in

C[si

, ai , . . . , an, x ]. Let us denote this family of polynomials by Ri,..., Rm.

5. Any C[ai,..., an, x]-linear relation for the family of polynomials Ri, ..., Rm transforms into a linear system of algebraic equations over the field of rational functions in the variables ai , . . . , an, x. Proposition 1 and Theorem 5 yield the existence of an at least one-dimensional C-vector space of solutions to this system of linear equations. Finding a basis in this space and clearing the denominators, we obtain the desired polynomial coefficients of the optimal annihilating operator for the initial algebraic function. □

Example 8. The linear space spanned by the roots of the algebraic equation

y5 + 2y4 — 3y3 + y2 + 5y + x = 0 (5)

(in a neighbourhood of a point where the discriminant of this equation does not vanish) coincides with the linear space of holomorphic solutions to the differential equation

(-43728190560 + 795819153 x - 53446888 x2 + 56028 x3)x x (-1585575 + 71982 x + 281583 x2 + 81342 x3 + 3125 x4) y(5) + +15(-650327879439783 - 5747872136026563x - 2400588229818366 x2--91559102743545 x3 - 304019551433 x4 - 131338505212 x5 + 128397500 x6) y(4) + +60(-1821690090417321 - 1560609625036728 x - 98711280942848 x2 + +721492325057x3 - 103787727624x4 + 91045500x5) y(3) + +180(-282046871305467 - 38794189010031 x + 478890241959 x2--28458003540 x3 + 21944300 x4) y''+

+720(-1756652589603 + 23053844253 x - 812236372 x2 + 522928 x3) y' = 0.

The following example provides a fundamental system of solutions to a fifth-order linear differential equation with polynomial coefficients.

Example 9. For any a e C a basis in the space of holomorphic solutions to the differential equation

((256/5) a5x3 + 625x4)y(5) + (384 a5x2 + 6875 x3)y(4) +

+ (624 a5x + 19500 x2)y(3) + (168 a5 + 14100 x)y'' + 1344y' = 0

in a neighbourhood of a generic point x e C is given by the roots of the algebraic equation

y5 + ay4 + x = 0.

3. Software, hardware and examples

Most of the examples in this paper were computed by means of a Mathematica 7.0 package developed by the authors and run on an Intel Core(TM) Duo CPU clocked at 2.00GHz. The bottleneck of the algorithm is computing the syzygy module of an ideal in a ring of polynomials in several variables. In some cases, we have used Singular for this.

Example 10. Generic quintic. One of the goals of the research presented in this paper was to provide a computationally efficient extension of the results of Section 5 in [11] beyond the class of algebraic equations with elementary solutions. In this example, we demonstrate the efficiency of the described approach by means of the generic monic quintic

y5 + a4y4 + a3 y3 + a2y2 + aiy + x = 0. (6)

Computing the annihilating operator for the solutions of this equation has turned out to be a task of considerable computational complexity. The full output of the algorithm is a vector of five polynomials with 4306 monomials in total and is too large to display. The degrees of these polynomials with respect to the variables ai, . . . , a4, x are 15, 20, 21, 22, 23. The leading coefficient of the annihilating operator has degree 7 with respect to x and splits into the product of two factors. One of them is the discriminant of (6) while the other is a polynomial of total degree 15 with 264 terms. The largest of the numeric coefficients in the annihilating operator for the generic monic quintic equals 2739594525000.

Example 11. A monic tetranomial with generic coefficients. By Theorem 5 the determination of a solution to the algebraic equation

y6 + ay2 + by + x = 0 (7)

equals five. The roots of (7) at a generic point x e C span the space of holomorphic solutions of the following fifth order linear differential operator with polynomial coefficients:

(-255664128 a10 + 395740000 a5b4 + 1599609375 b8 + 148780800 a6b2x + 2859609375 ab6x--499654656 a7x2 - 1573425000 a2b4x2 + 1051704000 a3b2x3 + 16796160 a4x4)x x (256 a5b2 +3125b6 - 1024a6x - 22500ab4x + 43200a2b2x2 - 13824a3x3 - 46656 x5)d^r +

+ (916300234752a16 + 18677130035200a11b4 - 38094525000000a6b8 - 134905517578125 ab12--77437887971328 a12b2 x + 107910691200000 a7b6x + 332702753906250 a2b10x+ +37877629059072 a13x2 - 406052352000 a8b4x2 +552267618750000 a3b8x2--128020162314240a9b2x3 - 727448202000000 a4b6x3 + 267464667561984 a10x4 + +43693344000000a5b4x4 - 1306049062500000b8x4 - 221398918963200a6b2x5--2201395927500000 ab6x5 + 359825022517248 a7x6 + 1137850610400000 a2b4x6--711490376448000a3b2x7 - 10579162152960 a4x8) d^r +

+30(-3264411795456 a12b2 + 5653930240000 a7b6 + 24016248046875 a2b10+ +3132022849536 a13x - 5896377011200 a8b4x - 145456875000 a3b8x--2678330105856 a9b2x2 - 31473123300000 a4b6x2 +38750783864832 a10x3--41250841920000a5b4x3 - 221475515625000 b8x3 - 21927996518400 a6b2x4--344319609375000 ab6 x4 + 52068310990848 a7x5 + 163366013160000 a2b4x5--95796142732800a3b2x6 - 1311148560384 a4x7) £3-

-120(-441539395584 a13 + 907546908800 a8b4 + 3520234921875 a3b8--122881784832 a9b2x + 2644039462500 a4b6x - 14126136238080 a10x2 + +20651647680000 a5b4x2 +88754326171875 b8x2 +4397772787200 a6b2 x3 + +122488079765625 ab6x3 - 16700798121984 a7x4 - 51772358925000 a2b4x4 + +28447624699200 a3b2x5 +352185242112 a4x6) £2 -

-2520(4704614400a9b2 +4079375000a4b6 - 219268374528a10x+ +350501380000 a5b4x + 1464693750000 b8x + 5043513600 a6 b2x2 + +1670568609375 ab6x2 - 196541448192 a7x3 - 597319515000 a2b4x3+ +304660958400 a3b2x4 + 3302125056 a4x5) £ +

+5040(-2372960256 a10 + 4421840000 a5b4 + 182 355 4 6 875 b8 - 1113523200 a6b2x+ +14120437500 ab6x - 1226244096 a7x2 - 3581820000 a2b4x2 + +1587859200 a3b2x3 + 13436928 a4x4).

Recall that the Newton polytope of a multivariate Laurent polynomial

f (xi ,...,!„)=

ai

xn) / ^ ca x1

supported in a finite set A is defined to be the convex hull of A. In what follows, we will not make any difference between two polytopes that only differ by a translation with respect to a real vector.

It turns out to be convenient to associate a single integer convex polytope with the optimal differential operator defined by an algebraic curve rather than consider the Newton polytope of each of its polynomial coefficients individually. Throughout the rest of the paper, we will be using the following definition.

Definition 12. By the Newton polytope of the optimal differential operator defined by a generic algebraic curve we will mean the convex hull of the exponents of all of the monomials that are present in its coefficients.

The following theorem describes the structure of the Newton polytope of the optimal differential operator for an algebraic curve. The idea of its proof has been suggested by L. Matusevich to whom the authors are very grateful.

Theorem 13. Let ^^(a1,..., an,x)d^ be the optimal annihilating operator for the algebraic

d _

fc=£

function defined by the relation P(x, y) := ym + a1ymi + ... + anymn + x = 0. Then the exponent vector (v1,..., vn+1) of any of the monomials that appear in (a1,..., an, x) lies in the

hyperplane , . , .

(m — m-!)«! + ... + (m — mn)vn + mvn+1 = const.

Proof. Recall that any germ of the algebraic equation y(a0,... ,an+1) defined by a0ym + a1 ymi + ... + anymn + an+1 = 0 satisfies the A-hypegeometric system of partial differential equations HA(P), where p = ( —1,0) and the matrix A is given by

A = ( 1 11 ... 11 '

y 0 mn mn-1 ... m1 m This yields, in particular, that the function y(a0,..., an+1) is A-homogeneous.

To simplify the notation, we denote x = an+1 and introduce the grading on the ring of ordinary differential operators C [a1,..., an, x, dX] by setting deg a = m — mi; for i = 1,..., n, deg a0 = 0 and degx = — deg (dx) = m. The annihilating ideal of the function y(a0,..., an,x) is homogeneous with respect to this grading. It follows from Proposition 1 and Theorem 2 that this ideal is principal. In fact, it is generated by the optimal linear differential annihilating operator for the function y(a0,..., an, x) with its order being equal to the determination of this function. Therefore, the optimal annihilating operator is also homogeneous with respect to the introduced grading.

Observe that since deg a0 = 0, passing over to the monic equation (that is, setting a0 = 1) has no effect on the grading. Thus the restriction of the optimal operator to ao = 1 is still homogeneous with respect to the grading defined above. Applying this homogeneous linear operator to the homogeneous function y(a1,..., an, x) which lies in its kernel, we conclude, that every coefficient of the optimal operator is a homogeneous polynomial with respect to the introduced grading.

Denoting by d^ the degree of the polynomial pl(a1,... ,an,x), with respect to the grading defined above, we conclude that the degree of (a1,..., an, x) equals d0 + k ■ m. This is exactly the conclusion of the theorem. □

Intensive experiments suggest that there is an intrinsic relation between the two extreme coefficients in the optimal annihilating differential operator for an algebraic function. As we have seen in several examples before, the leading coefficient in the annihilating operator is typically given by the product of the discriminant of the defining algebraic equation and some other factor which has no apparent relation to the initial algebraic equation. The following conjecture summarizes the results of our computer experiments with the structure of the Newton polytope of this polynomial factor.

d

Conjecture 14. Let ^^(a1,..., an, x)d^ be the optimal annihilating operator for the alge-

fc=£

braic function defined by the relation P(x,y) := ym + a1ymi + ... + anymn + x = 0. Denote by D(a1,..., an, x) the discriminant of P(x, y) computed with respect to y. Then the polynomials

pd(a1,. .., an, x)/D(a1,. .., an, x) and pl(a1,. .., an, x)

consist of monomials with the same exponent vectors. In particular, they contain equally many monomials and have equal Newton polytopes.

The following table illustrates Conjecture 14. It gives the linear ordinary differential operator whose solution space is spanned by the branches of an implicitly defined algebraic function y = y(x), the order of this operator and the multidegree of its leading coefficient with respect to x and the parameters of equation listed in lexicographic order.

Table 1. Computation times for annihilating operators and their properties

Algebraic curve The annihilating operator for y = y(x) Ord Leading coeff. Comput. time (sec.)

y4 + ay3 + x = 0 (27a4x2 - 256x3)dX + 4x(27a4 - 416x)dX +60(a4 - 36x)dX - 360dx 4 (2,4) 0.374

y4 + ay3 + by2 + x = 0 x(14b3 - 4a2b2 + 8bx - 3a2x)(16b4 - 4a2b3- 128b2x + 144a2bx - 27a4x + 256x2)dX +... + 120(74b3 - 21a2b2 + 24bx - 9a2x)dx 4 (4,6,7) 1.03

Algebraic curve The annihilating operator for y = y(x) Ord Leading coeff. Comput. time

(45c2 + 14b3 - 47abc - 4a2b2 + 12a3c + 8bx

y4 + ay3 + by2 + cy + x = 0 -3a2x)(27c4 + ... - 256c3)dX + ... + 120(243c2 + 74b3 - 249abc - 21a2b2 +63a3c + 24bx - 9a2x)dx 4 (4,7,7,6) 3.011

y5 + ay + x = 0 (256a5 + 3125x4)dX + 31250x3dX + 73125x2dX + 31875xdx - 1155 4 (4,5) 2.012

(51200b6 - 15930a4b3 - 2187a8 + 68000ab4x

- 1350a5bx - 26250a2b2x2)(256b5-

y5 + ay2 + by + x = 0 27a4b2 - 1600ab3x + 108a5x + 2250a2bx2 +3125x4)dX + ... +120(492800b6 - 139995a4b3 - 16038a8+ 222000ab4x + 8100a5bx - 39375a2b2x2) 4 (6,13,11) 5.008

y5 + ay3 + by2 + cy +x = 0 (102400c6 + ... - 2500a3bx3) (256c5 - 27b4c2 +144ab2c3 + ... + 3125x4)dX + ... - 120(985600c6 + ... - 625a3bx3) 4 (7,15, 13,11) 31.668

y5 + ay4 + x = 0 (265a5 x3 + 3125x4 )d£ + 5x2 (384a5 + 6875x)dX + 780x(4a5 + 125x)dX +60(14a5 + 1175x)dX + 6720dx 5 (4,5) 10.436

y5 + ay4 + by3 + x = 0 (1680ab9 + ... + 56a9x2)(108b5x2-27a2b4x2 + 2250b2x3 - 1600a3bx3+ 256a5x3 + 3125x4)dX + ... + 1680(20160ab9 + ... + 224a9x2)dx 5 (6,15,14) 38.563

y5 + ay4 + by3+ cy2 + x = 0 (160380c8 + ... - 56a9x3)(108c5x+ 16b3c3x + ... + 3125x4)dX +... + 1680(2779920c8 - 1242720b3c6 + ... -224a9x3)dx 5 (7,16, 14,13) 279.242

y5 + ay4 + by3+ cy2 + dy + x = 0 (4928000d6 + ... + 56a9x3)(256d5 - 27c4d2 +144bc2d3 + ... + 3125x4)dX + ... + 1680(9011200d6 + ... + 224a9x3)dx 5 (7,17,15, 13,11) 3038.47

y6 + ay3 + by2 + (6398437500c10 + ... + 67184640a2bc2x5) (3125c6 + 256b5 c2 + ... 5 (10,18, 799.427

cy + x = 0 -46656x5)dX + ... + 5040 (72942187500c10 + ... + 13436928a2bc2x5) 17,16)

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Многогранник Ньютона оптимального зануляющего оператора, связанного с алгебраической кривой

Виталий А. Красиков Тимур М. Садыков

В статье изучается линейный дифференциальный оператор с полниномиальными коэффициентами, чье пространство голоморфных решений порождается всеми ветвями алгебраической функции, заданной алгебраическим уравнением с коэффициентами общего положения. Основной результат состоит в описании коэффициентов этого оператора в терминах их многогранников Ньютона.

Ключевые слова: йлгебраическая функция, минимальный дифференциальный оператор, многогранник Ньютона.