[Paper Review] On the minimum of a positive polynomial over the standard simplex

Gabriela Jeronimo, Daniel Perrucci|ArXiv.org|Jun 23, 2009

Polynomial and algebraic computation10 references4 citations

TL;DR

This paper presents a new explicit lower bound for the minimum value of a positive polynomial with integer coefficients over the standard simplex in $ℝ^k$, using a deformation-based critical point method that avoids artificial degree growth. The key contribution is a doubly exponential bound in terms of the number of variables $k$, degree $d$, and coefficient bitsize $ au$, improving upon prior results by incorporating coefficient size and degree more precisely via algebraic geometry and polynomial system analysis.

ABSTRACT

We present a new positive lower bound for the minimum value taken by a polynomial P with integer coefficients in k variables over the standard simplex of R^k, assuming that P is positive on the simplex. This bound depends only on the number of variables, the degree and the bitsize of the coefficients of P and improves all previous bounds for arbitrary polynomials which are positive over the simplex.

Motivation & Objective

• To establish a tighter, explicit lower bound for the minimum value of a positive polynomial over the standard simplex in $ℝ^k$.
• To address the challenge of certifying positivity of polynomials over bounded domains, particularly the standard simplex, by providing an a priori lower bound.
• To improve upon existing bounds by incorporating the bitsize $τ$ of coefficients and the degree $d$ more effectively than previous methods.
• To develop a method that remains effective even in degenerate cases where traditional approaches fail due to coefficient growth or complex critical point structures.

Proposed method

• A deformation technique is applied to the polynomial $P$ by introducing a perturbation $F(t,X) = P(X) + tQ(X)$, where $Q(X) = \sum_{i=1}^k \frac{1}{d+1}X_i^{d+1}$, to analyze critical points via algebraic geometry.
• The critical points of $F(t,X)$ are studied through the variety $V(F_1,\dots,F_k)$ in $\mathbb{A}^{1} \times \mathbb{A}^{k}$, decomposed into components to isolate those relevant to the minimum at $t=0$.
• The method avoids sum-of-squares constructions by working directly with the polynomial system, preventing artificial degree inflation.
• Upper bounds on coefficients of the characteristic polynomial of a multiplication map in the quotient algebra are computed to estimate critical values without explicit root computation.
• A recursive argument is used to handle boundary minima by reducing the problem to lower-dimensional simplices, with adjusted coefficient size bounds.
• The final bound is derived by combining interior and boundary estimates, using induction on $k$ and careful comparison of exponential terms.

Experimental results

Research questions

• RQ1 What is the tightest possible explicit lower bound for the minimum of a positive polynomial with integer coefficients over the standard simplex, depending only on $k$, $d$, and $\tau$?
• RQ2 Can a method be developed that avoids the degree blowup from sum-of-squares representations while still certifying positivity via critical point analysis?
• RQ3 How does the inclusion of coefficient bitsize $\tau$ affect the tightness of the lower bound in the doubly exponential regime?
• RQ4 Is the doubly exponential dependence on $d$ and $\tau$ unavoidable, or can it be improved under certain structural assumptions?
• RQ5 Can the bound be made uniform across degenerate cases where critical points are not isolated or have high multiplicity?

Key findings

• The paper establishes a new lower bound: $\min_{\Delta_k} P \geq 2^{-(\tau+1)d^{k+1}} d^{-(k+1)d^k} \binom{d+k}{k+1}^{-d^k(d-1)}$, which improves upon all previous bounds for arbitrary positive polynomials over the simplex.
• The bound is doubly exponential in $d$ and $\tau$, and the paper shows this complexity is unavoidable via a constructive example achieving $O(2^{-\tau (d/2)^k})$ minimum.
• By using a deformation-based approach instead of sum-of-squares, the method avoids artificial degree growth and maintains tighter coefficient estimates.
• The bound is simplified to $\min_{\Delta_k} P \geq 2^{-(\tau+1)d^{k+1}} d^{-(k+1)d^{k+1}}$ using $\binom{d+k}{k+1} \leq d^{k+1}$, which is still tight in the asymptotic regime.
• The recursive treatment of boundary minima accounts for coefficient size inflation when reducing to lower-dimensional simplices, with a refined bound involving $\tau + 1 + d\log k$.
• The proof relies on induction on $k$, with a detailed inequality comparison showing that the new bound dominates all previous ones in the general case.