Newton polytopes and algebraic hypergeometric series

Newton Polytopes and Witness Sets

We present two algorithms that compute the Newton polytope of a polynomial f defining a hypersurface H in Cn using numerical computation. The first algorithm assumes that we may only compute values of f—this may occur if f is given as a straightline program, as a determinant, or as an oracle. The second algorithm assumes that H is represented numerically via a witness set. That is, it computes ...

Solving algebraic equations in terms of A-hypergeometric series

The roots of the general equation of degree n satisfy an A-hypergeometric system of diierential 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?1 triangulations of the Newton segment. This works over any eld whose characteristic is relatively prime to the lengths of the segments in the triangulation. ...

Elimination Theory and Newton Polytopes

Algebraic A-hypergeometric functions

We formulate and prove a combinatorial criterion to decide if an A-hypergeometric system of differential equations has a full set of algebraic solutions or not. This criterion generalises the so-called interlacing criterion in the case of hypergeometric functions of one variable.

Rational Parametrizations, Intersection Theory and Newton Polytopes

The study of the Newton polytope of a parametric hypersurface is currently receiving a lot of attention both because of its computational interest and its connections with Tropical Geometry, Singularity Theory, Intersection Theory and Combinatorics. We introduce the problem and survey the recent progress on it, with emphasis in the case of curves.