Über die lösung algebraischer gleichungssysteme durch hypergeometrische funktionen

Mayr, Karl

doi:10.1007/bf01708007

Cited by 11 publications

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“…, k m , d. It is not hard to see that the powers ρ s (x), s ∈ Z, of the roots of f (x; t), viewed as functions of the coefficients, are algebraic solutions of the A-hypergeometric system with exponent (0, −s). This fact was observed by Mayr [17] who constructed series expansions for these functions. These have more recently been refined by Sturmfels [22].…”

Section: Introductionmentioning

confidence: 55%

The 𝒜-hypergeometric system associated with a monomial curve

Cattani¹,

DʼAndrea²,

Dickenstein³

1999

Duke Math. J.

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Introduction. In this paper we make a detailed analysis of the Ꮽ-hypergeometric system (or GKZ system) associated with a monomial curve and integral, hence resonant, exponents. We describe all rational solutions and show in Theorem 1.10 that they are, in fact, Laurent polynomials. We also show that for any exponent there are at most two linearly independent Laurent solutions and that the upper bound is reached if and only if the curve is not arithmetically Cohen-Macaulay. We then construct, for all integral parameters, a basis of local solutions in terms of the roots of the generic univariate polynomial (0.5) associated with Ꮽ. We also determine in Theorem 3.7 the holonomic rank r(α) for all α ∈ Z 2 and show that d ≤ r(α) ≤ d +1, where d is the degree of the curve. Moreover, the value d +1 is attained only for those exponents α for which there are two linearly independent rational solutions, and, therefore, r(α) = d for all α if and only if the curve is arithmetically Cohen-Macaulay.In order to place these results in their appropriate context, we recall the definition of the Ꮽ-hypergeometric systems. These were introduced in a series of papers in the mid-1980s by the Gel fand school, particularly Gel fand, Kapranov, and Zelevinsky (see [7] and [9] and the references therein). Let Ꮽ = {ν 1 , . . . , ν r } ⊂ Z n+1 be a finite subset that spans the lattice Z n+1 . Suppose, moreover, that there exists a vector λ = (λ 0 , . . . , λ n ) ∈ Q n+1 such that λ, ν j = 1 for all j = 1, . . . , r, that is, the set Ꮽ lies in a rational hyperplane. Let Ꮽ also denote the (n + 1) × r matrix whose columns are the vectors ν j . Let ᏸ ⊂ Z r be the sublattice of elements v ∈ Z r such that Ꮽ · v = 0. Given α ∈ C n+1 , the Ꮽ-hypergeometric system with exponent (or parameter) α is The Ꮽ-hypergeometric system is holonomic (with regular singularities) and, consequently, the number of linearly independent solutions at a generic point is finite (see [7]). Let r(α) denote the holonomic rank of the system, that is, the dimension of the space of local solutions at a generic point in C r . If we drop the assumption that Ꮽ lies in a hyperplane, then the regular singularities property is lost, but, as Adolphson [2] has shown, the system remains holonomic. The singular locus is described by the zeroes of the principal Ꮽ-determinant (see [11]). We set R := C[ξ 1 , . . . , ξ r ]/Ᏽ Ꮽ , where Ᏽ Ꮽ is the toric idealWhen n = 1, we can assume without loss of generality that is then a rational solution with the same exponent. Similarly, one can show that the local residuesgive algebraic solutions with exponent (−a, −b) and, again, the total sum of residues is a rational solution.THE Ꮽ-HYPERGEOMETRIC SYSTEM OF A MONOMIAL CURVE 181In §1 we describe explicitly all rational solutions of the Ꮽ-hypergeometric system associated with a monomial curve. Since for Ꮽ as in (0.4) the principal Ꮽ-determinant factors into powers of x 0 , x d , and the discriminant (f ), we know a priori what the possible denominators of a rational solution may be. However, we show in T...

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Section: Introductionmentioning

confidence: 55%

The 𝒜-hypergeometric system associated with a monomial curve

Cattani¹,

DʼAndrea²,

Dickenstein³

1999

Duke Math. J.

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“…It follows as corollary of Thrm. 18 and well known properties of symbolic roots of univariate algebraic equations [Stu00, May36,Bir27] that polynomials used to specify PDEs for F ≃S , F ⊂ ∼ S and F ⊃ ∼ S include in their congruence class a hypergeometric functions of product of linear form which is symmetric with respect to permutations of the edge variables. In particular it follows that neither polynomials P ≃S (x), P ⊂ ∼ S (x) and P ⊃ ∼ S (x) can be expressed as a sum over fewer then S 2 ( n 2 ) /Gn product of linear forms symmetric with respect to permutation of the edge variables.…”

Section: Andmentioning

confidence: 99%

On Partial Differential Encodings of Boolean Functions

Gnang¹,

Xu²

2020

Preprint

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In a epoch-making book George Bool [Boo54] laid the foundation for what has come to be known as the Boolean algebra. The Boolean algebra undoubtedly, serves as one of the pillars to the digital computing revolution. Interestingly, enough George Bool also initiated the branch of mathematics knowns as invariant theory. There is a recognition [GIM + 19, GMQ16, Aar16, Gro20] of a rich interplay relating these seemingly independent branches which happen to have been both pioneered by George Bool. Invariant theory approaches emphasize important consequence which stem from the presence of symmetries or lack thereof when analyzing of the complexity of Boolean functions. The importance of symmetry in the analysis of Boolean function was already well know to pioneers of the field including among many others Shannon, Pólya and Redfield [Sha49,Pol40,Pol37,Red27]. In this work we pursue a partial derivative incarnation of Turing machines. Turing machines were introduced by Alan Turing[Tur36] and serve as the second pillar to the digital computing revolution. The differential operator approaches to invariants and complexity theory is very old, it traces its roots to the work of early pioneers of invariant theory. Most notably to Arthur Cayley and James Joseph Sylvester [Cay89, Syl52] who instigated the analysis of differential operator used to construct invariants of group actions. This particular framework has come to be known as Cayley's Ω process. In complexity theory, differential operators were investigated in the context of arithmetic complexity by Baur and Strassen who analyzed the arithmetic complexity of computing partial derivative [BS83] More recently, Cornelius Brand and Kevin Pratt in [BP20] were able to match the runtime of the fastest known deterministic algorithm for detecting subgraphs of bounded pathwidth using a Method of partial derivatives. We also refer the reader to the excellent survey of recent uses of partial derivatives in arithmetic complexity theory [CKW11, SY10]. The importance of partial differential operators is also reinforced by the central role they play in Deep Learning through back propagation routines. The present work ties together rather strongly, aspects of arithmetic complexity to Boolean complexity. Fortunately, recent depth reduction results[VS81, AV08, Raz13, GKKS16, Hya79] reduce investigations of arithmetic circuit lower bounds to the analysis of depth 3 arithmetic circuits. These results motivate our emphasize on low depth arithmetic circuits. Finally, we argue that better understanding of arithmetic circuit complexity is central to Boolean circuit complexity and not merely a first step. We do this by showing that Boolean circuit complexity upperbounds the Kolomogorov complexity of certain partial derivative incarnations of Turing machines.

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“…In the next theorem, we use the A-hypergeometric partial differential resolvents of Mayr [5] and Gelfand et al [1] and their differential consequences to demonstrate that all compositions of the derivations D and Θ can be expressed as linear combinations over Q{e} of a certain set, ∂ ,w , of partial derivatives of distinct weights. Although explicit formulae have been determined, for instance in [6, Theorem 59, page 109], for the expression of the operator D m Θ i as a linear combination of the partial derivatives, ∂ ,w , the computation of these formulae is not enlightening and involves no original ideas.…”

Section: Partial Differential Resolventsmentioning

confidence: 99%

A partial factorization of the powersum formula

Nahay¹

2004

International Journal of Mathematics and Mathematical Sciences

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For any univariate polynomial P whose coefficients lie in an ordinary differential field Ã°ÂÂ”Â½ of characteristic zero, and for any constant indeterminate ÃŽÂ±, there exists a nonunique nonzero linear ordinary differential operator Ã¢Â„Âœ of finite order such that the ÃŽÂ±th power of each root of P is a solution of Ã¢Â„ÂœzÃŽÂ±=0, and the coefficient functions of Ã¢Â„Âœ all lie in the differential ring generated by the coefficients of P and the integers Ã¢Â„Â¤. We call Ã¢Â„Âœ an ÃŽÂ±-resolvent of P. The author's powersum formula yields one particular ÃŽÂ±-resolvent. However, this formula yields extremely large polynomials in the coefficients of P and their derivatives. We will use the A-hypergeometric linear partial differential equations of Mayr and Gelfand to find a particular factor of some terms of this ÃŽÂ±-resolvent. We will then demonstrate this factorization on an ÃŽÂ±-resolvent for quadratic and cubic polynomials

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Über die lösung algebraischer gleichungssysteme durch hypergeometrische funktionen

Cited by 11 publications

References 0 publications

The 𝒜-hypergeometric system associated with a monomial curve

The 𝒜-hypergeometric system associated with a monomial curve

On Partial Differential Encodings of Boolean Functions

A partial factorization of the powersum formula

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