2015/01/01 by Dmitrii V. Pasechnik, DV Pasechnik, Pasechnik, Dmitrii V. +1 · 2 voices
Computer Science · Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Advanced Optimization Algorithms Research #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Classical Analysis and ODEs (math.CA) #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Optimization and Control (math.OC) #Polynomial and algebraic computation #math.AG #math.CA #math.OC
paper · pdf · doi:10.48550/arxiv.1511.03473
openalex publication_date 2015/01/01 · arxiv published 2015/11/11 · openalex created_date 2025/10/10 · arxiv updated 2026/03/18 · openalex updated_date 2026/07/28
For any 4-variate quartic form f≥ 0 (i.e. f nonnegative, homogeneous polynomial of degree 4 with real coefficients) there exist quadratic forms q and q' so that qq'f is a sum of squares (s.o.s.) of quartics, by reducing to the case of f=au2+2bu+c with a, b, c 3-variate forms of degrees 2, 3, 4, respectively, and invoking on its discriminant Δ=ac-b2 a theorem by Hilbert (1893) asserting that for any ternary sextic h≥ 0 there exists a quadric q'' so that q''h is s.o.s. of quartics. Towards deciding whether just one q always suffices to make qf a s.o.s, we give explicit examples of non-s.o.s. f=au2+2bu+c≥ 0 with non-s.o.s. Δ. However, in all these examples af are s.o.s. That is, the straightforward s.o.s. decomposition via Hilbert (1893) need not be the best possible. While it remains open whether one q always suffices (and we conjecture that q=a suffices), we describe how the existence of such q is related to particular types of s.o.s. decompositions for Δ.