Newton polytopes and witness sets

TL;DR

This paper introduces two numerical algorithms for computing the Newton polytope of a hypersurface, one using only polynomial evaluations and the other leveraging witness sets, with applications to invariants like the L"uroth invariant.

Contribution

It presents novel algorithms for Newton polytope computation based solely on numerical data, applicable to complex hypersurfaces and invariants.

Findings

First algorithm works with polynomial evaluations only.

Second algorithm uses witness sets for hypersurface representation.

Applied to compute the Newton polytope of the L"uroth invariant.

Abstract

We present two algorithms that compute the Newton polytope of a polynomial defining a hypersurface H in C^n using numerical computation. The first algorithm assumes that we may only compute values of f - this may occur if f is given as a straight-line 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 the Newton polytope of H using only the ability to compute numerical representatives of its intersections with lines. Such witness set representations are readily obtained when H is the image of a map or is a discriminant. We use the second algorithm to compute a face of the Newton polytope of the L\"uroth invariant, as well as its restriction to that face.