[Paper Review] Signature pairs of positive polynomials

Jennifer Halfpap, Jiř́í Lebl|arXiv (Cornell University)|Nov 5, 2012

Mathematical functions and polynomials11 references7 citations

TL;DR

This paper establishes effective bounds on the ratio of negative to positive signature pairs for bihomogeneous polynomials that become squared norms after multiplication by $\|z\|^{2d}$, proving sharp bounds for $d=1$ and optimal-order bounds for $d>1$. It shows that the classes $\Psi_d$ of such polynomials are strictly increasing with $d$, and provides explicit lower bounds on the ratio $N_-/N_+$ that grow as $C_n d^{n-1}$, confirming the correct asymptotic order in $d$ for fixed $n$. The results refine Quillen's theorem by quantifying the minimal $d$ needed for positivity to imply a Hermitian sum of squares structure.

ABSTRACT

A well-known theorem of Quillen says that if $r(z,\bar{z})$ is a bihomogeneous polynomial on ${\mathbb{C}}^n$ positive on the sphere, then there exists $d$ such that $r(z,\bar{z}){\lVert z Vert}^{2d}$ is a squared norm. We obtain effective bounds relating this $d$ to the signature of $r$. We obtain the sharp bound for $d=1$, and for $d > 1$ we obtain a bound that is of the correct order as a function of $d$ for fixed $n$. The current work adds to an extensive literature on positivity classes for real polynomials. The classes $Ψ_d$ of polynomials for which $r(z,\bar{z}){\lVert z Vert}^{2d}$ is a squared norm interpolate between polynomials positive on the sphere and those that are Hermitian sums of squares.

Motivation & Objective

• To quantify the minimal $d$ such that a bihomogeneous polynomial positive on the sphere becomes a Hermitian sum of squares after multiplying by $\|z\|^{2d}$, extending Quillen's theorem with effective bounds.
• To determine the sharp upper bound on the ratio $N_-/N_+$ of negative to positive signature pairs for polynomials in $\Psi_d$, particularly for $d=1$ and $d>1$.
• To show that the classes $\Psi_d$ are strictly increasing: $\Psi_0 \subsetneq \Psi_1 \subsetneq \Psi_2 \subsetneq \cdots$, by constructing explicit examples in $\Psi_{d+1} \setminus \Psi_d$.
• To establish the correct asymptotic order of the ratio $N_-/N_+$ as a function of $d$ for fixed $n$, proving it grows as $\Theta(d^{n-1})$.

Proposed method

• Define the positivity classes $\Psi_d$ as bihomogeneous polynomials $r$ such that $r(z,\bar{z})\|z\|^{2d}$ is a Hermitian sum of squares.
• Use the signature pair $(N_+, N_-)$ of a real polynomial to characterize its decomposition into positive and negative Hermitian squares.
• Apply inductive constructions in the coefficient space of $p \cdot \ell^d$, where $\ell$ is a linear form, to control the sign pattern of coefficients in the product.
• Construct polynomials $p$ with controlled negative and positive coefficient patterns across layers (indexed by powers of $x_2$) to ensure $p \cdot \ell^d$ has non-negative coefficients for $d$-th power, while maximizing $N_-/N_+$.
• Use a recursive strategy: for $d>1$, alternate positive and negative layers in the coefficient diagram, placing positive coefficients only on every $\nu$-th layer to improve the ratio $N_-/N_+$.
• Prove the lower bound $C_n d^{n-1}$ via induction on $n$, using the recurrence $C_n = C_{n-1}/2^{n-1}$ with base case $C_2 = 1/2$.

Experimental results

Research questions

• RQ1 What is the sharp upper bound on the ratio $N_-/N_+$ for polynomials in $\Psi_1$?
• RQ2 For $d > 1$, what is the correct asymptotic order of the maximal $N_-/N_+$ ratio as a function of $d$ for fixed $n$?
• RQ3 Can the classes $\Psi_d$ be strictly separated, i.e., is $\Psi_d \subsetneq \Psi_{d+1}$ for all $d \geq 0$?
• RQ4 How can one effectively construct polynomials in $\Psi_{d+1} \setminus \Psi_d$ to demonstrate strict inclusion?
• RQ5 What is the minimal $d$ such that a positive polynomial on the sphere lies in $\Psi_d$, and how does this depend on its signature?

Key findings

• For $d=1$, the bound $N_-/N_+ < n-1$ is sharp, and for any $\varepsilon > 0$, there exists a polynomial with $N_-/N_+ \geq n-1 - \varepsilon$.
• For $d > 1$, the bound $N_-/N_+ \leq \binom{n-1+d}{d} - 1$ holds, and this bound is of the correct order $\Theta(d^{n-1})$ for fixed $n$.
• There exists a constant $C_n > 0$ such that for each $d$, there is a polynomial in $\Psi_d$ with $N_-/N_+ \geq C_n d^{n-1}$, proving the bound is optimal in order.
• The classes $\Psi_d$ are strictly increasing: $\Psi_0 \subsetneq \Psi_1 \subsetneq \cdots$, as shown by constructing polynomials in $\Psi_{d+1} \setminus \Psi_d$ for all $d$.
• The asymptotic growth rate of $C_n$ is $C_n = 1/2^{n(n-1)/2}$, derived from the recurrence $C_n = C_{n-1}/2^{n-1}$ with $C_2 = 1/2$.
• The construction method using layered coefficient patterns and alternating positive layers achieves the optimal $d^{n-1}$ growth rate for the ratio $N_-/N_+$.