The DeLiVerMATH project - Text analysis in mathematics

A high-quality content analysis is essential for retrieval functionalities but the manual extraction of key phrases and classification is expensive. Natural language processing provides a framework to automatize the process. Here, a machine-based approach for the content analysis of mathematical texts is described. A prototype for key phrase extraction and classification of mathematical texts is presented

Hasse invariants and mod pp solutions of AA-hypergeometric systems

Igusa noted that the Hasse invariant of the Legendre family of elliptic curves over a finite field of odd characteristic is a solution mod $p$ of a Gaussian hypergeometric equation. We show that any family of exponential sums over a finite field has a Hasse invariant which is a sum of products of mod $p$ solutions of $A$-hypergeometric systems.Comment: 22 page

AA-hypergeometric systems that come from geometry

We establish some connections between nonresonant $A$-hypergeometric systems and de Rham-type complexes. This allows us to determine which of these $A$-hypergeometric systems "come from geometry."Comment: 10 page

Distinguished-root formulas for generalized Calabi-Yau hypersurfaces

By a "generalized Calabi-Yau hypersurface" we mean a hypersurface in ${\mathbb P}^n$ of degree $d$ dividing $n+1$. The zeta function of a generic such hypersurface has a reciprocal root distinguished by minimal $p$-divisibility. We study the $p$-adic variation of that distinguished root in a family and show that it equals the product of an appropriate power of $p$ times a product of special values of a certain $p$-adic analytic function ${\mathcal F}$. That function ${\mathcal F}$ is the $p$-adic analytic continuation of the ratio $F(\Lambda)/F(\Lambda^p)$, where $F(\Lambda)$ is a solution of the $A$-hypergeometric system of differential equations corresponding to the Picard-Fuchs equation of the family.Comment: 33 page

On the zeta function of a projective complete intersection

We compute a basis for the p-adic Dwork cohomology of a smooth complete intersection in projective space over a finite field and use it to give p-adic estimates for the action of Frobenius on this cohomology. In particular, we prove that the Newton polygon of the characteristic polynomial of Frobenius lies on or above the associated Hodge polygon. This result was first proved by B. Mazur using crystalline cohomology.Comment: 24 pages, no figure

Newton polytopes and algebraic hypergeometric series

Let $X$ be the family of hypersurfaces in the odd-dimensional torus ${\mathbb T}^{2n+1}$ defined by a Laurent polynomial $f$ with fixed exponents and variable coefficients. We show that if $n\Delta$, the dilation of the Newton polytope $\Delta$ of $f$ by the factor $n$, contains no interior lattice points, then the Picard-Fuchs equation of $W_{2n}H^{2n}_{\rm DR}(X)$ has a full set of algebraic solutions (where $W_\bullet$ denotes the weight filtration on de Rham cohomology). We also describe a procedure for finding solutions of these Picard-Fuchs equations.Comment: With an appendix by Nicholas Kat

Exponential sums on A^n, III

We give two applications of our earlier work "Exponential sums on A^n, II" (math.AG/9909009). We compute the p-adic cohomology of certain exponential sums on A^n involving a polynomial whose homogeneous component of highest degree defines a projective hypersurface with at worst weighted homogeneous isolated singularities. This study was motivated by recent work of Garcia (Exponential sums and singular hypersurfaces, Manuscripta Math., v. 97 (1998), pp. 45-58). We also compute the p-adic cohomology of certain exponential sums on A^n whose degree is divisible by the characteristic.Comment: 15 pages, LaTeX2

Dwork cohomology, de Rham cohomology, and hypergeometric functions

In the 1960s, Dwork developed a p-adic cohomology theory of de Rham type for varieties over finite fields, based on a trace formula for the action of a Frobenius operator on certain spaces of p-adic analytic functions. One can consider a purely algebraic analogue of Dwork's theory for varieties over a field of characteristic zero and ask what is the connection between this theory and ordinary de Rham cohomology. N. Katz showed that Dwork cohomology coincides with the primitive part of de Rham cohomology for smooth projective hypersurfaces, but the exact relationship for varieties of higher codimension has been an open question. In this article, we settle the case of smooth affine complete intersections.Comment: 20 page

On logarithmic solutions of A-hypergeometric systems

For an $A$-hypergeometric system with parameter $\beta$, a vector $v$ with minimal negative support satisfying $Av = \beta$ gives rise to a logarithm-free series solution. We find conditions on $v$ analogous to `minimal negative support' that guarantee the existence of logarithmic solutions of the system and we give explicit formulas for those solutions. Although we do not study in general the question of when these logarithmic solutions lie in a Nilsson ring, we do examine the $A$-hypergeometric systems corresponding to the Picard-Fuchs equations of certain families of complete intersections and we state a conjecture regarding the integrality of the associated mirror maps.Comment: 23 page

AA-hypergeometric series associated to a lattice polytope with a unique interior lattice point

We associate to lattice points a_0,a_1,...,a_N in Z^n an A-hypergeometric series \Phi(\lambda) with integer coefficients. If a_0 is the unique interior lattice point of the convex hull of a_1,...,a_N, then for every prime p\neq 2 the ratio \Phi(\lambda)/\Phi(\lambda^p) has a p-adic analytic continuation to a closed unit polydisk minus a neighborhood of a hypersurface.Comment: 12 page