2010/08/27 by Martín Avendaño, Térésa Krick, Avendano, Martin +1
Computer Science · Mathematics · #11S05 #13F30 #Advanced Differential Equations and Dynamical Systems #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation
paper · doi:10.48550/arxiv.1008.4808
openalex publication_date 2010/08/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
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)<=t2 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)<=(t2-t+1)(pf-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)(pf-1), which gives the sharp estimation Bm(2,K)=3(pf-1) for trinomials when p>2+e.