Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Stubborn Polynomials

Lorenzo Baldi, Grigoriy Blekherman, Khazhgali Kozhasov, Daniel Plaumann, Bruce Reznick, Rainer Sinn

TL;DR

This work develops a comprehensive theory of stubborn polynomials, i.e., nonnegative forms on a real variety $X$ for which no odd power is a sum of squares, linking algebraic geometry with real algebraic certificates. It establishes a tight link between stubbornness and real zeros on smooth curves, proving that on irreducible totally real curves stubbornness coincides with real-rootedness for nonnegative non-SOS forms, and that stubborn forms emerge in high degree on curves of positive genus while rational curves generally lack them. It extends to singular and reducible curves via $2$-torsion in generalized Jacobians, and develops a lifting framework from curves (notably elliptic normal curves) to higher-dimensional varieties, yielding new stubborn forms in ambient spaces and clarifying when lifting preserves nonnegativity. The ternary sextic case yields a sharp characterization: a nonnegative sextic not SOS is stubborn iff its real delta invariant satisfies $ ext{delta}^{ ext{R}}(F)                                      $9$. The results illuminate the geometry of nonnegativity certificates, provide explicit constructions and counterexamples, and open avenues (lifting, real-torsion, divisor conditions) for discovering new stubborn forms in higher dimensions with controlled real zeroes.

Abstract

The relationship between nonnegative polynomials and sums of squares is a classical topic in real algebraic geometry. We study \emph{stubborn polynomials} $f$ on a real variety $X$, which are polynomials nonnegative on $X$, such that no odd power of $f$ is a sum of squares. Previously, stubborn polynomials were studied only in the globally nonnegative case, with results restricted to polynomials nonnegative on $\mathbb{P}^2$. We fully characterize stubborn polynomials on smooth curves, showing that a polynomial on a smooth totally real curve is stubborn if and only if all of its zeros are real. This implies that there exist smooth curves with no stubborn polynomials in low degree, while stubborn polynomials must exist in sufficiently high degrees on curves with positive genus. We explore the much more delicate situation with singular and reducible curves. While being real-rooted always implies being stubborn, there also exist singular curves with no stubborn polynomials at all. We then analyze the case of ternary sextics, i.e.,~polynomials of degree $6$ on $\mathbb{P}^2$. We prove the Conjecture of Blekherman, Kozhasov, and Reznick that a nonnegative ternary sextic is stubborn if and only if its real delta-invariant is at least 9. To analyze this case, we develop results for lifting stubborn polynomials from curves to higher dimensional varieties and use the theory of weak Del Pezzo surfaces. We complement these results with structural properties of stubborn polynomials and present many explicit examples.

Stubborn Polynomials

TL;DR

This work develops a comprehensive theory of stubborn polynomials, i.e., nonnegative forms on a real variety for which no odd power is a sum of squares, linking algebraic geometry with real algebraic certificates. It establishes a tight link between stubbornness and real zeros on smooth curves, proving that on irreducible totally real curves stubbornness coincides with real-rootedness for nonnegative non-SOS forms, and that stubborn forms emerge in high degree on curves of positive genus while rational curves generally lack them. It extends to singular and reducible curves via -torsion in generalized Jacobians, and develops a lifting framework from curves (notably elliptic normal curves) to higher-dimensional varieties, yielding new stubborn forms in ambient spaces and clarifying when lifting preserves nonnegativity. The ternary sextic case yields a sharp characterization: a nonnegative sextic not SOS is stubborn iff its real delta invariant satisfies 9$. The results illuminate the geometry of nonnegativity certificates, provide explicit constructions and counterexamples, and open avenues (lifting, real-torsion, divisor conditions) for discovering new stubborn forms in higher dimensions with controlled real zeroes.

Abstract

The relationship between nonnegative polynomials and sums of squares is a classical topic in real algebraic geometry. We study \emph{stubborn polynomials} on a real variety , which are polynomials nonnegative on , such that no odd power of is a sum of squares. Previously, stubborn polynomials were studied only in the globally nonnegative case, with results restricted to polynomials nonnegative on . We fully characterize stubborn polynomials on smooth curves, showing that a polynomial on a smooth totally real curve is stubborn if and only if all of its zeros are real. This implies that there exist smooth curves with no stubborn polynomials in low degree, while stubborn polynomials must exist in sufficiently high degrees on curves with positive genus. We explore the much more delicate situation with singular and reducible curves. While being real-rooted always implies being stubborn, there also exist singular curves with no stubborn polynomials at all. We then analyze the case of ternary sextics, i.e.,~polynomials of degree on . We prove the Conjecture of Blekherman, Kozhasov, and Reznick that a nonnegative ternary sextic is stubborn if and only if its real delta-invariant is at least 9. To analyze this case, we develop results for lifting stubborn polynomials from curves to higher dimensional varieties and use the theory of weak Del Pezzo surfaces. We complement these results with structural properties of stubborn polynomials and present many explicit examples.
Paper Structure (14 sections, 23 theorems, 26 equations, 5 figures)

This paper contains 14 sections, 23 theorems, 26 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Stubborn polynomials on varieties
  3. Existence of stubborn polynomials on curves
  4. Stubborn polynomials on plane cubics
  5. Singular curves and positive $2$-torsion points
  6. Curves with no stubborn polynomials
  7. Smooth rational curves
  8. Smooth plane curves with no real-rooted quadrics
  9. No real-rooted polynomials via Descartes' rule of signs.
  10. Nonnegative lifting
  11. Ternary sextics and weak Del Pezzo surfaces
  12. Sextics with few zeroes are not stubborn
  13. Sextics with many zeroes are stubborn
  14. Questions and Conjectures

Key Result

Theorem 1.1

Let $X \subset \mathbb{P}^n$ be an irreducible, totally real, projective $d$-normal curve. Let $F \in {\mathbb R}[X]_{2d}$ be nonnegative on $X({\mathbb R})$ but not a sum of squares. Consider the following conditions: Then $(1) \implies (2)$, and if $X$ is smooth, then $(2) \implies (1)$. Furthermore, if $X$ is smooth of genus $g\ge 1$ and $d$ is large enough, there exist stubborn forms in ${\ma

Figures (5)

  • Figure 1: A stubborn quadratic on an irreducible cubic.
  • Figure 2: A stubborn quadric for a circle with a secant line.
  • Figure 3: A stubborn quartic for a circle with a tangent or passing line.
  • Figure 4: A stubborn quartic on three lines through a point.
  • Figure 5: The Edge quartic and nonnegative octic built from its bitangents.

Theorems & Definitions (56)

  • Theorem 1.1: see \ref{['thm:many_zeroes_implies_stubborn', 'thm:stubborn_implies_many_zeroes', 'thm:large degree']}
  • Theorem 1.2: see Theorems \ref{['thm:stubbornSextics']} and \ref{['thm:delta_geq_9']}
  • Theorem 1.3: see \ref{['thm:general_lifting']} and \ref{['cor:ext']}
  • Theorem 1.5: see \ref{['cor:ext_elliptic']}
  • Example 1.6
  • Definition 2.1
  • Theorem 2.2
  • proof
  • Corollary 2.3
  • Definition 2.4
  • ...and 46 more