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An efficient sum of squares nonnegativity certificate for quaternary quartic

Dmitrii V. Pasechnik

Abstract

For any 4-variate quartic form $f\geq 0$ (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 $f=au^2+2bu+c$ with $a$, $b$, $c$ $3$-variate forms of degrees 2, 3, 4, respectively, and invoking on its discriminant $Δ=ac-b^2$ a theorem by Hilbert (1893) asserting that for any ternary sextic $h\geq 0$ 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. $f=au^2+2bu+c\geq 0$ 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 $Δ$.

An efficient sum of squares nonnegativity certificate for quaternary quartic

Abstract

For any 4-variate quartic form (i.e. nonnegative, homogeneous polynomial of degree with real coefficients) there exist quadratic forms and so that is a sum of squares (s.o.s.) of quartics, by reducing to the case of with , , -variate forms of degrees 2, 3, 4, respectively, and invoking on its discriminant a theorem by Hilbert (1893) asserting that for any ternary sextic there exists a quadric so that is s.o.s. of quartics. Towards deciding whether just one always suffices to make a s.o.s, we give explicit examples of non-s.o.s. with non-s.o.s. . However, in all these examples 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 always suffices (and we conjecture that suffices), we describe how the existence of such is related to particular types of s.o.s. decompositions for .

Paper Structure

This paper contains 10 sections, 11 theorems, 23 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Reduction to a real zero case
  3. On sharpness of the s.o.s. degree bound for 4-ary 4-ics
  4. Reducible discriminant
  5. Preliminaries
  6. Construction
  7. Towards s.o.s. for quaternary quartics
  8. Proof of Theorem \ref{['thm:sos34']}.
  9. An SDP for s.o.s. with a multiplier
  10. Code to check computations

Key Result

Theorem 2

Let $f\in P_{{4},{4}}$. Then $f=\frac{p}{qq'}$, where $p\in\Sigma_{{4},{8}}$ and $q,q'\in\Sigma_{{4},{2}}$.

Figures (3)

  • Figure 1: Computations for the proof of \ref{['thm:exDelta']}.
  • Figure 2: Computations to check that $af_*$ in \ref{['subsect:remarks']} is s.o.s.
  • Figure 3: Computations for the polynomial $f_*$ in \ref{['subsect:remarks']}.

Theorems & Definitions (18)

  • Remark 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Lemma 5
  • proof
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • proof
  • ...and 8 more