[Paper Review] Nonnegative Polynomials and Sums of Squares

Grigoriy Blekherman|arXiv (Cornell University)|Oct 18, 2010

Advanced Optimization Algorithms Research12 references4 citations

TL;DR

This paper identifies the fundamental geometric and algebraic reason why some nonnegative homogeneous polynomials are not sums of squares in the smallest cases: ternary sextics (3,6) and quaternary quartics (4,4). It shows that Cayley-Bacharach relations—linear dependencies among polynomials—generate all linear inequalities that separate the cone of nonnegative polynomials from the cone of sums of squares, providing a complete characterization of these distinctions via intersection points of curves and explicit linear functionals that certify non-sos status.

ABSTRACT

In the smallest cases where there exist nonnegative polynomials that are not sums of squares we present a complete explanation of this distinction. The fundamental reason that the cone of sums of squares is strictly contained in the cone of nonnegative polynomials is that polynomials of degree $d$ satisfy certain linear relations, known as the Cayley-Bacharach relations, which are not satisfied by polynomials of full degree 2d. For any nonnegative polynomial that is not a sum of squares we can write down a linear inequality coming from a Cayley-Bacharach relation that certifies this fact. We also characterize strictly positive sums of squares that lie on the boundary of the cone of sums of squares and extreme rays of the cone dual to the cone of sums of squares

Motivation & Objective

• To explain why nonnegative polynomials are not always sums of squares in the smallest cases where this distinction occurs: (3,6) and (4,4).
• To identify the complete set of linear inequalities that separate the cone of nonnegative forms from the cone of sums of squares in these cases.
• To show that these separating inequalities arise exclusively from Cayley-Bacharach relations among intersection points of curves.
• To characterize extreme rays of the dual cone of sums of squares using real and complex intersection points of forms.

Proposed method

• Uses homogenization to work with forms instead of general polynomials, focusing on the cases (3,6) and (4,4).
• Applies Cayley-Bacharach relations—linear dependencies among polynomials of degree d that do not hold for full-degree 2d polynomials—to derive separating linear inequalities.
• Constructs linear functionals ℓ(f) = ∑μif(zi) based on evaluation at intersection points γi of curves (cubics in (3,6), quadrics in (4,4)), with complex conjugate pairs allowed.
• Defines a real evaluation map Eℝ: Hn,d → ℝ^s that maps forms to their values at real and complex-conjugate points, and identifies the image as a hyperplane L.
• Analyzes positive semidefinite quadratic forms Qℓ on Hn,d induced by these evaluations, showing they are extreme rays of the dual cone Σ∗n,2d.
• Completes the square in the quadratic form to reduce the problem to a real hyperplane and proves that extreme rays correspond either to point evaluations or to forms with a unique projective zero in the kernel.

Experimental results

Research questions

• RQ1 What is the fundamental algebraic and geometric reason that nonnegative polynomials are not sums of squares in the (3,6) and (4,4) cases?
• RQ2 Do all linear inequalities separating the cone of nonnegative forms from the cone of sums of squares arise from Cayley-Bacharuch relations in these cases?
• RQ3 Can extreme rays of the dual cone of sums of squares be fully characterized using intersection points of curves, including complex conjugate pairs?
• RQ4 Is the presence of complex intersection points necessary for extreme rays, or can all such rays be realized via fully real intersections?

Key findings

• In the (3,6) case, any nonnegative form that is not a sum of squares is certified by a linear functional ℓ(f) = ∑μif(zi) based on 9 intersection points of two real cubics, with at most two complex points.
• In the (4,4) case, such certification arises from 8 intersection points of three real quadrics, with at most two complex points.
• All linear inequalities that separate Pn,2d from Σn,2d in these cases are generated by Cayley-Bacharach relations, which are absent in full-degree 2d polynomials.
• Extreme rays of the dual cone Σ∗n,2d that are not point evaluations correspond to quadratic forms Qℓ with a unique projective zero in the kernel, arising from complex-conjugate pairs of intersection points.
• The kernel of an extreme ray Qℓ not corresponding to point evaluation has the property that any transverse intersection of n−1 forms in it contains at most one complex conjugate pair of zeros.
• The paper conjectures that such extreme rays can always be realized via fully real intersections, implying complex pairs are not essential for the characterization.