Patterns of dependence among powers of polynomials

2001/06/08 by Bruce Reznick, Reznick, Bruce
Mathematics · #11D41 #11E76 #11P05 #14Q15 #15A99 #30D35 (Secondary) #32H25 (Primary) #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics #Number Theory (math.NT) #Rings and Algebras (math.RA) #math.AG #math.CV #math.NT #math.RA #msc:11D41 #msc:11E76 #msc:11P05 #msc:14Q15 #msc:15A99 #msc:30D35 #msc:32H25

paper · pdf · doi:10.48550/arxiv.math/0106060

Submitted to Contemp. Math., for the Proceedings of the March 2001 DIMACS workshop on Algorithmic and Quantitative Aspects of Real Algebraic Geometry in Mathematics and Computer Science. The preprint is 25 pp. and a few typos have been corrected from a circulated version

Abstract

Let F = f₁,...,f_r be a family of polynomials and let the ticket of F, T(F), denote the set of integers m so that f_j^m is linearly dependent. We show that |T(F)| ≤ (r-1)(r-2)/2 and present many concrete examples, including one with r=6 and T(F) = 1,2,3,4,8,14.

Related

• Linear independence of powers for polynomials
• On multiplicative dependence between elements of polynomial orbits
• Detecting linear dependence on a semiabelian variety
• On the linear independence of shifted powers
• Polynomial patterns in subsets of large finite fields of low characteristic