Sharp bounds for the number of roots of univariate fewnomials

TL;DR

This paper establishes new upper bounds for the maximum number of roots of univariate polynomials with few terms over various fields, using Vandermonde determinants and field extension properties.

Contribution

It introduces a unified approach to bound the roots of sparse polynomials over local fields and finite extensions, improving existing estimates with explicit formulas.

Findings

Proves Bm(t,L) <= t^2 Bm(t,K) for local fields with non-archimedean valuation.

Provides an upper bound Bm(t,K) <= (t^2 - t + 1)(p^f - 1) for finite extensions of Qp.

Establishes a lower bound Bm(t,K) >= (2t - 1)(p^f - 1) for odd p, with exact value for t=2.

Abstract

Let K be a field and t>=0. Denote by Bm(t,K) the maximum number of non-zero roots in K, counted with multiplicities, of a non-zero polynomial in K[x] with at most t+1 monomial terms. We prove, using an unified approach based on Vandermonde determinants, that Bm(t,L)<=t^2 Bm(t,K) for any local field L with a non-archimedean valuation v such that v(n)=0 for all non-zero integer n and residue field K, and that Bm(t,K)<=(t^2-t+1)(p^f-1) for any finite extension K/Qp with residual class degree f and ramification index e, assuming that p>t+e. For any finite extension K/Qp, for p odd, we also show the lower bound Bm(t,K)>=(2t-1)(p^f-1), which gives the sharp estimation Bm(2,K)=3(p^f-1) for trinomials when p>2+e.