Solving Algebraic Equations in Terms of A-Hypergeometric Series. Bernd Sturmfels. Department of Mathematics. University of California

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1 Solving Algebraic Equations in Terms of A-Hypergeometric Series Bernd Sturmfels Department of Mathematics University of California Berkeley, CA 9472, U.S.A. Abstract The roots of the general equation of degree n satisfy an A-hypergeometric system of dierential equations in the sense of Gel'fand, Kapranov and Zelevinsky. We construct the n distinct A-hypergeometric series solutions for each of the 2 n? triangulations of the Newton segment. This works over any eld whose characteristic is relatively prime to the lengths of the segments in the triangulation.. Solving the Quintic A classical problem in mathematics is to nd a formula for the roots of the general equation of degree n in terms of its n coecients. While there are formulas in terms of radicals for n 4, Galois theory teaches us that no such formula exists for the general quintic a x a 4 x 4 x 3 x 2 a x a = : (:) An alternative approach is to expand the roots into fractional power series (or Puiseux series). In 77 Johann Lambert expressed the roots of the trinomial equation x p x r as a Gauss hypergeometric function in the parameter r. Series expansions of more general algebraic functions were subsequently given by Euler, Chebyshev and Eisenstein, among others. The poster \Solving the Quintic with Mathematica" [2] gives a nice introduction to these classical techniques and underlines their relevance for symbolic computation. The state of the art in the rst half of our century appears in works of Richard Birkeland [2] and Karl Mayr []. They proved that the roots are multivariate hypergeometric functions (in the sense of Horn) in all of the coecients and they gave series expansions for the roots and their powers. The purpose of this note is to rene the these results. Our point of departure is the fact that the roots satisfy the A-hypergeometric dierential equations introduced by Gel'fand, Kapranov and Zelevinsky [6],[7]. Here A denotes the conguration of n equidistant points on the ane line. It follows from recent work

2 of McDonald [] that there are 2 n? distinct complete sets of series solutions, one for each of the 2 n? triangulations of A. We shall construct these series solutions explicitly. Let us illustrate our general construction for the example of the quintic (.). Here the set A has 6 distinct triangulations. The nest triangulation divides A into ve segments of unit length. The coarsest triangulation of A is just a single segment of length. For the nest triangulation we get the following expressions for the ve roots of (.): X ;? =? a a ; X2;? =? a a a ; X3;? =? a ; X 4;? =? a 4 ; X;? =? a 4 a a 4 : Each bracket represents a power series having the monomial in the bracket as its rst term: a a a a2 a3 a 4 a4 a = a a a2 = a a2 2 =? a a 3 = a 4? a 4 = a 4 a? a3 a 4? a3 a 4 a 4 2? a a 4 3 a2 3a 4 2 a3 2 a? 3 a a a a a 3 a 4 a4 a 4 a 2 a3 a3 3 a 2 a2 2a 4 3? 3 a 4 4? a4 a 6 a4 a a 2? a3 2a a 4 3? a a 4 4? a a a 6? a4 a 4 a a3 2 4 a 3 4 a a 4 Note that the last bracket is just a single Laurent monomial. The other four brackets can easily be written as an explicit sum over N4. For instance, ai? a i a a = X i;j;k;l (?) 2i3j4kl (2i3j4kl)! i! j! k! l! (i2j3k4l )! ai2j3k4l a i 2 aj 3 ak 4 al i3j4kl Each coecient appearing in one of these series is integral. Therefore our ve series solutions of the general quintic are characteristic-free. They work over any base eld. The situation is dierent for the coarsest triangulation of A. Here we must assume that the characteristic is dierent from. The ve series solutions of (.) are a = X ; = a = 2 a = = 3 2 = = 4 a = a 4=? a 4 a

3 where runs over the ve roots of the equation =?. The brackets denote the series a = a a = = = = = a = a 4= = a= a = = = = a? 2 = = = = a = a 4= a a 4 a 4= a 6=? 2? 3 = a 7=? 2? a 7= =? 3? a a 6= a 4=? 2 a 4= a 6= a 4 = a 7= a 4 = a 8= 4 a = a 9= a 9= a 6= 4 = = a a 4 a 7= a 8= a = a 4= a 4= a = a a 8= a 7= a 3 a 7= a 8= a a 4 a 6= a 9= Each of these four series can be expressed as an explicit sum over the lattice points in a 4-dimensional polyhedron. The general formula will be presented in Theorem 3.2 below. 2. The Roots are A-hypergeometric Our problem is to compute the roots of the general equation of degree n, f(x) = a a x x 2 : : : a n? x n? a n x n : (2:) Each root of f(x) is an algebraic function in the indeterminate coecients: X = X(a ; a ; ; : : : ; a n? ; a n ): Proposition 2.. (Karl Mayr [, p. 284]) The roots of the general equation of degree n satisfy the following system of linear partial dierential 2 j l whenever i j = k l; (2:2) nx i= i =?X and nx i= i = : (2:3) The system (2.2)-(2.3) is a special instance of the class of A-hypergeometric dierential equations introduced by Gel'fand, Kapranov and? Zelevinsky [],[7]. Namely, (2.2)-(2.3) is? the A-hypergeometric system with parameters associated with the integer matrix A := 2 3 n? n : (2:4) The column vectors of A are homogeneous coordinates of n equidistant points on the line. Their convex hull is a line segment: it is the Newton polytope of f(x). 3

4 The Euler-type equations (2.3) follow readily from the homogeneity relations X(a ; ta ; t 2 ; : : : ; t n? a n? ; t n a n ) = t X(a ; a ; ; : : : ; a n? ; a n ); X(ta ; ta ; t ; : : : ; ta n? ; ta n ) = X(a ; a ; ; : : : ; a n? ; a n ): The equations (2.2) appeared in Karl Mayr's 937 paper [, equation (2) on page 284]. I shall present two proofs dierent proofs. The rst one was shown to me in the spring of 992 by Jean-Luc Brylinski. See [3] for an appearence of (2.2) in dierential geometry. Brylinsky's proof of (2.2): It uses implicit dierentiation and works over any base eld. We consider the rst derivative f (x) = P n i= ia ix i? and the second derivative f (x) = P n i=2 i(i? )a ix i?2. Note that f (X) 6=, since a ; : : : ; a n are indeterminates. Dierentiating the dening identity P n i= a ix(a ; a ; : : : ; a n ) i = with respect to a j, we get X j j = : (2:) We next j with respect to the indeterminate a i 2 j X i f (X) (X) X j f (X)?2 jx f (X)? : i Using (2.) and the i =? f (X) f (X) Xi ix i?, we can rewrite (2.6) 2 j =?f (X)X ij f (X)?3 (i j)x ij? f (X)?2 : (2:7) The expression (2.7) depends only on the sum of indices i j. This proves (2.2). A complex analysis proof of (2.2): Suppose we are working over the eld of complex numbers C. Consider the logarithmic derivative [log(f(x))] = f (x)=f(x). We view it as a rational function in a ; a ; : : : ; a n and dierentiate with respect to these 2? [log(f(x))]?x ij = j f 2 (x) This shows that [log(f(x))] satises the quadratic A-hypergeometric equations (2.2). Proposition 2. follows by dierentiating under the integral sign in Cauchy's formula X = 2i where? is a suciently small loop in the complex plane. 4 Z? zf (z) dz; (2:8) f(z)

5 3. Series Expansions For any rational number u and any integer v we abbreviate (u; v) := 8 >< >: if v =, u(u? )(u? 2) (u v ) if v <, if u is a negative integer and u?v, otherwise. (u)(u2)(uv) Let L denote the integer kernel of A. This is the (n? )-dimensional sublattice of Z n spanned by fe i?? 2e i e i : i = ; : : : ; n?g. Consider any monomial a u au au n n with rational exponents in the coecients of f(x). We dene the formal power series a u au au n n := X ny (v ;:::;v n )2L i=? (ui ; v i ) a u i v i i : (3:) This series satises the quadratic dierential equations (2.2), and it satises the linear equations (2.3) with their right hand sides?x and replaced by X and 2 X. Lemm.. ([, Lemma ]) The series a u au n n is a formal solution of the A-hypergeometric system with parameters? 2 = A (u ; u ; : : : ; u n ) T. Gel'fand, Kapranov and Zelevinsky constructed a complete set of series solutions for each regular triangulations of the set A. We shall adapt their general construction to our special case. Extra care must be taken, however, because our equations (2.3) do not satisfy the non-resonance hypothesis which is necessary for [, Theorem 3] to hold. We write ; ; : : : ; n for the points in our conguration A in (2.4). It has 2 n? triangulations, all of which are regular. Each triangulation is indexed by a subset I of f; : : : ; n? g. Writing the complementary subset as f; ; : : : ; ngni = f = i < i < i 2 < < i r? < i r = ng, the triangulation of A indexed by I consists of the r segments [i ; i ]; [i ; i 2 ]; : : : ; [i r? ; i r ]. See [8, Sections 7.3.A and 2.2.A] for details. If r = n (resp. r = ) then this is the nest (resp. coarsest) triangulation referred to in Section. We are now prepared to present the main result of this note. In the remainder of Section 3 we shall be working over the eld of complex numbers C. Fix any of the 2 n? triangulations of A. For j = ; : : : ; r we write :=?? for the length of the j-th segment in that triangulation. Clearly, d d 2 d r = n. Let = (?) = be any of the -th roots of?. We dene the A-hypergeometric series X j; := a? a? d X j k=2 k a ij? k? a k?? a? k aij?? a ij? a If j = then the expression? a appears in the rightmost summand. We dene it to be zero. Note that by varying j and we have dened n distinct series in total.

6 Theorem 3.2. The n series X j; are roots of the general equation of order n, that is, f(x j; ) =. There exists a constant M such that all n series X j; converge whenever ja ij? j ij?k ja ij j k?? M ja k j for all j r and k 62 f? ; g: (3:2) Proof: Consider the open convex cone C = w 2 R n : (?k)w ij? (k?? )w ij < w k for j = ; : : :; r; k 62 f? ; g : This is the outer normal cone to the secondary polytope (A) at the vertex corresponding to the triangulation I of A (cf. [8, Theorem 2.2.2]). Equivalently, the cone C consists of all vectors w = (w ; w ; : : : ; w n ) which induce the triangulation in question. Let U be the region in the coecient space C n dened by the inequalities (3.2) for M. There exists a vector V 2 R n such that (log(ja j); : : : ; log(ja n j))? V 2 C for all (a ; : : : ; a n ) in U. Let H be the space? of all complex-valued functions on U which are? A-hypergeometric with parameters. The A-hypergeometric system is holonomic of rank n and its singular locus, the discriminantal locus of f, is disjoint from U (see []). Hence H is a complex vector space of dimension at most n. We shall identify n linearly independent elements in H, which will imply that H has dimension exactly n. It follows from [, Proposition 2] that the series a u au n n dened in (3.) converges in U provided at most two of the exponents u i are non-integers. Each of the summands in the denition of X j; has this property. Therefore X j; converges in U. Using Lemm., we conclude that the n series X j; lie in the vector space H. We x an vector w = (w ; : : : ; w n ) in C such that all coordinates w i are integers and such that no two of the lines spanned by pairs f(w i ; i); (w j ; j)g in R 2 are parallel. The weight of a monomial a i a i a i n n (with rational exponents) is dened to be w i w i w n i n. We replace the input equation by its toric deformation f t (x) = a t w a t w x t w 2 x 2 : : : a n? t w n? x n? a n t w n x n : (3:3) We shall study the n roots as an algebraic function of t. For t close to the origin the n roots split into r groups, one for each segment [? ; ], for j = ; : : : ; r. (This is a special case of the multivariate construction in [9, x3].) The roots in the j-th group possess a Puiseux expansion of the form j;(t) = a? a? t (w ij??w ij ) higher terms in t; where satises =?. We shall prove that j; () = X j; for all (a ; : : : ; a n ) 2 U. To this end we rst determine the second lowest term in the series j; (t). After modifying the weight vector w by an ane transformation w i 7! w i, which does not alter the 6

7 structure of the series, we may assume that w ij = w ij? =, and there is a unique index r with w r =, and w l > for l 2 f; ; : : : ; ngnf ;? ; rg. An explicit calculation reveals j;(t) = a? a? r?? a r a (r?)=? a (??r?)= t higher terms in t: (3:4) By varying w within the cone C we can arrange that the role of r is played by any index in the set? f; ; : : : ; ngnfi ; i 2 ; : : : ; i r g [ f? ; 2 g: (3:) Note that the cardinality of this set is at least?. Consider the linear map that extracts from a series in a ; : : : ; a n all those terms which are lowest or second lowest with respect to the grading dened by some w 2 C. Call this map T. Consider the image of a root j; () under T. This is a polynomial (with fractional exponents) having at least distinct terms, one for the lowest term in (3.4) and at least? for the distinct values of r coming from (3.). The coecients of these terms are distinct powers of. These considerations imply that the images of the roots j; () under T are linearly independent. Therefore the roots themselves are linearly independent? over C. Moreover, they all satisfy? the A-hypergeometric system with parameter, by Proposition 2.. We conclude that the space H is n-dimensional and the roots j; () form a basis for H. We wish to prove j; () = X j;. It suces to show T ( j; ()) = T (X j; ) because the functional T dened above separates the space H. Equivalently, given any generic w in C, we must show that, in the w-grading, X j; has the same rst two terms as j;(). This is clear for the rst term, so we only need to look at the second term r?? a r a (r?)=? a (??r?)=. Let a v denote this monomial. Let k be the unique integer between and such that r is congruent to??k modulo. If k = then a v equals the second lowest term (with respect to w) of the series [a =? a?= ]. If 2 k? then a v equals the w-lowest term of the series indexed by k in the central sum in the denition of X j;. For k = there are two subcases: if r >? then a v equals the w-lowest term of the series indexed by k = in the center sum; if r <? then a v equals the w-lowest term of the rightmost series [a ij??=a ij?]. In each of these four cases a v is the second lowest term in X j;. This completes the proof of Theorem

8 4. Integrality Issues To prove the identity f(x j; ) = in Theorem 3.2 we used the complex numbers. But, a posteriori, there is no need to stick with an algebraically closed eld of characteristic zero. If K is any integral domain such that the equation =? has distinct solutions and the coecients of X j; are dened in K, then f(x j; ) = is a valid identity in a suitable fractional power series ring over K. The following result shows that K can be an algebraically closed eld of characteristic p, provided p is relatively prime to d d 2 d r. The proof of Theorem 4. using Hensel's Lemma was suggested to me by Hendrik Lenstra. Theorem 4.. All coecients of the series X j; lie in the ring Z[ ][]. Proof: The series j; (t) in (3.4) lies in the fractional power series ring R [[t ]], where R := Q[ ] a s : s 62 f? ; g [a? ; a ]: This holds because! is integral and each term of X j;xi = j; () lies in R. To prove Theorem 4., it suces to show that j; (t) is an element of the subring R 2 [[t ]], where R 2 := Z[ ][ ] a s : s 62 f? ; g [a? ; a ]: We shall apply Hensel's Lemma [, Theorem 7.3] to the integral domain R 2 [[t ]]. This domain is complete with respect to the principal ideal m := ht i, and it contains all coecients of the polynomial f t (x) in (3.3). The constant term of j; (t) lies in R 2 ; we? a? denote it by A := a. It is an \approximate root" in the sense that f t (A) 2 m and ft(a) is the sum of a unit in R 2 and an element of m. By Hensel's Lemma there exists a unique element B in R 2 [[t ]] such that ft (B) = and A? B 2 m. By repeating the uniqueness part of this application of Hensel's Lemma for the coecient domain R instead of R 2, we conclude that B must be equal to our Puiseux series j; (t). A case of special interest is the nest triangulation, where I = f; 2; : : : ; n?g, r = n, = for all j. Theorem 4. implies that in this case our construction is characteristic-free. In other words, the series solutions X j;? have integer coecients. These series are X j;? =? aj? a j aj?2 a j? for j = ; 2; : : : ; n (4:) a where j? a j is the sum over all Laurent monomials (?)? i : : :? : : : i n ai ai a?2 j?2 a? j? a j ai n n a j (4:2) 8

9 where i ; i ; : : : ; i n are non-negative integers satisfying the relations i i i 2 i 3?? i n = i 2i 2 3i 3 (j? )?? j (j ) ni n = : (4:3) For generating the series a j? a j on a computer it convenient to rewrite (4.3) as follows.? =?ji? (j? )i? (j? 2)i 2?? 2?2 22 (n? j)i n ; =?(j? )i? (j? 2)i??? (n? j )i n : These equations ensure that the multinomial coecient in (4.3) is divisible by?.. Multivariate Outlook Consider a system of n polynomial equations in n variables, where the terms in the i- th equation have their exponent vectors in a xed set A i Z n. (This is the meaning of \sparse" in [9]). By Bernstein's Theorem, the system has mixed volume many roots (X ; : : : ; X n ). Each coordinate X i is an algebraic function in all the coecients. It is natural to wonder whether X i satises the A-hypergeometric dierential equations where A is the conguration arising from the \Cayley trick" (cf. [7, 2.], [8, page 273]): A := A fe g [ A 2 fe 2 g [ [ A n fe n g Z 2n : The answer is no. The coordinates of the roots (X ; : : : ; X n ) are not A-hypergeometric. For example, let (X ; X 2 ) be the unique solution of the system of two linear equations Then we nd a a x x 2 = b b x b 2 x 2 = : Note also that Mayr's dierential equations (4) and (4) in [] are no longer binomial. The correct generalization of Proposition 2. to higher dimensions is the following. Let J(x ; : : : ; x n ) denote the Jacobian j ) of the given equations f = = f n =. Proposition.. For any integers u ; u 2 ; : : : ; u n the algebraic function X u Xu 2 2 Xu n n J(X ; X 2 ; : : : ; X n ) (:) satises the A-hypergeometric dierential equations arising from the Cayley trick. Over the eld of complex numbers, we can derive Proposition. from Theorem 2.7 in [7] using Cauchy's formula in several variables. An alternative algebraic proof follows from 9

10 the construction in [, x2]. The quantity in (.) should be thought of as a local residue on a toric variety [4]. The sum over all (mixed volume many) local residues is the global residue associated with the monomial x u xu n n relative to the given equations f = = f n =. The global residue is a rational function. It is of importance in elimination theory. We plan to extend the techniques in Section 3 to the setting of Proposition. in a subsequent joint work with E. Cattani and A. Dickenstein. The goal of that project is to develop new formulas and algorithms for computing local and global residues. Another interesting question is whether explicit Puiseux series expansions of (.) might be useful to improve the numerical component in the homotopy algorithm proposed in [9]. References [] A. Adolphson and S. Sperber: Dierential modules dened by systems of equations, Seminario Rendiconti di Universita di Padova, to appear. [2] R. Birkeland: Uber die Auosung algebraischer Gleichungen durch hypergeometrische Funktionen, Mathematische Zeitschrift 26 (927) 6{78. [3] J.-L. Brylinski: Radon transform and functionals on the space of curves, Manuscript, 993. [4] E. Cattani, D. Cox, A. Dickenstein: Residues in toric varieties, Manuscript, 99. [] D. Eisenbud: Commutative Algebra with a View Towards Algebraic Geometry, Graduate Texts in Mathematics, Springer Verlag, New York, 99. [6] I.M. Gel'fand, A.V. Zelevinsky and M.M. Kapranov: Hypergeometric functions and toral manifolds, Functional Analysis and its Applications 23 (989) 94{6. [7] I.M. Gel'fand, A.V. Zelevinsky and M.M. Kapranov: Generalized Euler integrals and A-hypergeometric functions, Advances in Mathematics (99) 84 2{27. [8] I.M. Gel'fand, A.V. Zelevinsky and M.M. Kapranov: Discriminants, Resultants, and Multidimensional Determinants, Birkhauser, Boston, 994. [9] B. Huber and B. Sturmfels: A polyhedral method for solving sparse polynomial systems, Mathematics of Computation 64 (99) 4{. [] K. Mayr: Uber die Auosung algebraischer Gleichungssysteme durch hypergeometrische Funktionen, Monatshefte fur Mathematik und Physik 4 (937) 28{33. [] J. McDonald: Fiber polytopes and fractional power series, Journal of Pure and Applied Algebra, to appear. [2] Solving the Quintic with Mathematica, poster distributed by Wolfram Research, 994.

(1.) For any subset P S we denote by L(P ) the abelian group of integral relations between elements of P, i.e. L(P ) := ker Z P! span Z P S S : For ea

(1.) For any subset P S we denote by L(P ) the abelian group of integral relations between elements of P, i.e. L(P ) := ker Z P! span Z P S S : For ea Torsion of dierentials on toric varieties Klaus Altmann Institut fur reine Mathematik, Humboldt-Universitat zu Berlin Ziegelstr. 13a, D-10099 Berlin, Germany. E-mail: altmann@mathematik.hu-berlin.de Abstract