2020/10/21 by Alexander Varchenko, Varchenko, Alexander · 1 citation
Computer Science · Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2010.11275
openalex publication_date 2020/10/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the KZ differential equations over mathbb C in the case, when\nthe hypergeometric solutions are one-dimensional integrals. We also consider\nthe same differential equations over a finite field mathbb Fp. We study the\npolynomial solutions of these differential equations over mathbb Fp,\nconstructed in a previous work joint with V. ,Schechtman and called the\n mathbb Fp-hypergeometric solutions.\n The dimension of the space of mathbb Fp-hypergeometric solutions depends\non the prime number p. We say that the KZ equations have ample reduction for\na prime p, if the dimension of the space of mathbb Fp-hypergeometric\nsolutions is maximal possible, that is, equal to the dimension of the space of\nsolutions of the corresponding KZ equations over mathbb C. Under the\nassumption of ample reduction, we prove a determinant formula for the matrix of\ncoordinates of basis mathbb Fp-hypergeometric solutions. The formula is\nanalogous to the corresponding formula for the determinant of the matrix of\ncoordinates of basis complex hypergeometric solutions, in which binomials\n(zi-zj)Mi+Mj are replaced with (zi-zj)Mi+Mj-p and the Euler\ngamma function \Γ(x) is replaced with a suitable mathbb Fp-analog\n\Γ mathbb Fp(x) defined on mathbb Fp.\n