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On Exact Reznick, Hilbert-Artin and Putinar's Representations

Victor Magron, Mohab Safey El Din

TL;DR

This work develops a hybrid numeric-symbolic framework for computing exact rational sums of squares decompositions of multivariate polynomials by combining semidefinite programming with symbolic absorption. The method starts from a perturbation $f_\varepsilon=f-\varepsilon\sum_{\alpha\in Q} X^{2\alpha}$ to place the polynomial in the interior of the SOS cone, obtains a numerical Gram decomposition via SDP, and then recovers an exact certificate with rational coefficients. It then extends this approach to obtain exact Reznick representations for positive definite forms, Hilbert-Artin representations for nonnegative polynomials, and Putinar representations for polynomials positive on compact semialgebraic sets, with explicit bit-complexity bounds that are singly exponential in the Newton polytope size. The authors implement these algorithms in Maple within the RealCertify suite and demonstrate practical performance and trade-offs against existing methods such as the rounding-projection approach, critical-point methods, and CAD. The results advance exact nonnegativity certificates in real algebraic geometry and open avenues for handling border cases and noncommutative variants in future work.

Abstract

We consider the problem of computing exact sums of squares (SOS) decompositions for certain classes of non-negative multivariate polynomials, relying on semidefinite programming (SDP) solvers. We provide a hybrid numeric-symbolic algorithm computing exact rational SOS decompositions with rational coefficients for polynomials lying in the interior of the SOS cone. The first step of this algorithm computes an approximate SOS decomposition for a perturbation of the input polynomial with an arbitrary-precision SDP solver. Next, an exact SOS decomposition is obtained thanks to the perturbation terms and a compensation phenomenon. We prove that bit complexity estimates on output size and runtime are both singly exponential in the cardinality of the Newton polytope (or doubly exponential in the number of variables). Next, we apply this algorithm to compute exact Reznick, Hilbert-Artin's representation and Putinar's representations respectively for positive definite forms and positive polynomials over basic compact semi-algebraic sets. We also report on practical experiments done with the implementation of these algorithms and existing alternatives such as the critical point method and cylindrical algebraic decomposition.

On Exact Reznick, Hilbert-Artin and Putinar's Representations

TL;DR

This work develops a hybrid numeric-symbolic framework for computing exact rational sums of squares decompositions of multivariate polynomials by combining semidefinite programming with symbolic absorption. The method starts from a perturbation to place the polynomial in the interior of the SOS cone, obtains a numerical Gram decomposition via SDP, and then recovers an exact certificate with rational coefficients. It then extends this approach to obtain exact Reznick representations for positive definite forms, Hilbert-Artin representations for nonnegative polynomials, and Putinar representations for polynomials positive on compact semialgebraic sets, with explicit bit-complexity bounds that are singly exponential in the Newton polytope size. The authors implement these algorithms in Maple within the RealCertify suite and demonstrate practical performance and trade-offs against existing methods such as the rounding-projection approach, critical-point methods, and CAD. The results advance exact nonnegativity certificates in real algebraic geometry and open avenues for handling border cases and noncommutative variants in future work.

Abstract

We consider the problem of computing exact sums of squares (SOS) decompositions for certain classes of non-negative multivariate polynomials, relying on semidefinite programming (SDP) solvers. We provide a hybrid numeric-symbolic algorithm computing exact rational SOS decompositions with rational coefficients for polynomials lying in the interior of the SOS cone. The first step of this algorithm computes an approximate SOS decomposition for a perturbation of the input polynomial with an arbitrary-precision SDP solver. Next, an exact SOS decomposition is obtained thanks to the perturbation terms and a compensation phenomenon. We prove that bit complexity estimates on output size and runtime are both singly exponential in the cardinality of the Newton polytope (or doubly exponential in the number of variables). Next, we apply this algorithm to compute exact Reznick, Hilbert-Artin's representation and Putinar's representations respectively for positive definite forms and positive polynomials over basic compact semi-algebraic sets. We also report on practical experiments done with the implementation of these algorithms and existing alternatives such as the critical point method and cylindrical algebraic decomposition.

Paper Structure

This paper contains 18 sections, 21 theorems, 80 equations, 4 tables, 7 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Exact SOS representations
  4. Algorithm intsos
  5. Correctness and bit size of the output
  6. Bit complexity analysis
  7. Comparison with the rounding-projection algorithm of Peyrl and Parrilo
  8. Exact Reznick and Hilbert-Artin's representations
  9. Exact Reznick's representations
  10. Exact Hilbert-Artin's representations
  11. Exact Putinar's representations
  12. Preliminary results
  13. Algorithm $\texttt{Putinarsos}$
  14. Bit complexity analysis
  15. Comparison with the rounding-projection algorithm of Peyrl and Parrilo
  16. ...and 3 more sections

Key Result

Theorem 1

Reznick78 Let $f \in \Sigma[X]$ with $f = \sum_{i=1}^r s_i^2$, $P := \mathop{\mathrm{\mathcal{C}}}\nolimits{(f)}$ and $Q := P/2 \cap \mathbb{N}^n$. Then for all $i=1,\dots,r$, $\mathop{\mathrm{\mathcal{C}}}\nolimits{(s_i)} \subseteq Q$.

Theorems & Definitions (39)

  • Theorem 1
  • Theorem 2
  • Lemma 3
  • Proposition 4
  • proof
  • Proposition 5
  • proof
  • Proposition 6
  • Remark 7
  • Example 8
  • ...and 29 more