2023/11/13 by Tomasz Kowalczyk, Kowalczyk, Tomasz, Julian Vill +1
Mathematics · #11P05 #14Q30 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #History and Theory of Mathematics
paper · pdf · doi:10.48550/arxiv.2311.07356
openalex publication_date 2023/11/13 · openalex created_date 2023/11/15 · openalex updated_date 2026/07/28
We show that the higher Pythagoras numbers for the polynomial ring are infinite p2s(K[x1,x2,…,xn])=∞ provided that K is a formally real field, n≥2 and s≥ 1. This almost fully solves an old question \cite[Problem 8]cldr1982. The remaining open cases are precisely n=1 and s>1. Moreover, we study in detail the cone of binary octics that are sums of fourth powers of quadratic forms. We determine its facial structure as well as its algebraic boundary. This can also be seen as sums of fourth powers of linear forms on the second Veronese of ℙ1. As a result, we disprove a conjecture of Reznick \cite[Conjecture 7.1]reznick2011 which states that if a binary form f is a sum of fourth powers, then it can be written as f=f12+f22 for some nonnegative forms f1,f2.