[Paper Review] Computing rational points in convex semi-algebraic sets and SOS decompositions

Mohab Safey El Din, Lihong Zhi|ArXiv.org|Oct 15, 2009

Polynomial and algebraic computation19 references3 citations

TL;DR

This paper presents a polynomial-time algorithm for computing rational points in convex semi-algebraic sets defined by integer polynomial inequalities, with bit complexity bounded by $\sigma^{O(1)}D^{O(k^3)}$, where $\sigma$ is the bit length of coefficients and $D$ the maximum degree. It applies this to decide whether a rational polynomial is a sum of squares (SOS) in $\mathbb{Q}[x_1,\ldots,x_n]$, providing a rational SOS decomposition when one exists, with coefficients of bit length $\tau D^{O(k^3)}$ for a degree-$2d$ polynomial with $\tau$-bit coefficients.

ABSTRACT

Let ${\cal P}=\{h_1, ..., h_s\}\subset \Z[Y_1, ..., Y_k]$, $D\geq °(h_i)$ for $1\leq i \leq s$, $σ$ bounding the bit length of the coefficients of the $h_i$'s, and $Φ$ be a quantifier-free ${\cal P}$-formula defining a convex semi-algebraic set. We design an algorithm returning a rational point in ${\cal S}$ if and only if ${\cal S}\cap \Q eq\emptyset$. It requires $σ^{\bigO(1)}D^{\bigO(k^3)}$ bit operations. If a rational point is outputted its coordinates have bit length dominated by $σD^{\bigO(k^3)}$. Using this result, we obtain a procedure deciding if a polynomial $f\in \Z[X_1, >..., X_n]$ is a sum of squares of polynomials in $\Q[X_1, ..., X_n]$. Denote by $d$ the degree of $f$, $τ$ the maximum bit length of the coefficients in $f$, $D={{n+d}\choose{n}}$ and $k\leq D(D+1)-{{n+2d}\choose{n}}$. This procedure requires $τ^{\bigO(1)}D^{\bigO(k^3)}$ bit operations and the coefficients of the outputted polynomials have bit length dominated by $τD^{\bigO(k^3)}$.

Motivation & Objective

• To design an algorithm that decides whether a convex semi-algebraic set defined over integers contains rational points.
• To compute a rational point in such a set when one exists, with bounded bit length for coordinates.
• To apply this to the problem of determining whether a rational polynomial is a sum of squares of rational polynomials.
• To provide a complexity analysis for both the rational point computation and the SOS decomposition procedures.

Proposed method

• The algorithm uses quantifier elimination over the reals via a procedure from Basu et al. (1996) to transform the semi-algebraic set into a quantifier-free form.
• It reduces the problem of rational point existence in a convex semi-algebraic set to solving a system of non-strict polynomial inequalities with integer coefficients.
• The method leverages semidefinite programming duality by representing the SOS condition as a positive semidefinite matrix constraint on coefficients.
• A rational parametrization of the solution set is used, and the average of real roots of a univariate polynomial yields a rational point when all roots are real.
• The algorithm computes a rational point by exploiting symmetry and algebraic structure in the solution variety of the defining inequalities.
• Bit complexity is analyzed by bounding the size of intermediate coefficients and the number of arithmetic operations in terms of $D$, $k$, and $\sigma$.

Experimental results

Research questions

• RQ1 Can a convex semi-algebraic set defined by integer polynomial inequalities contain a rational point, and if so, can it be computed efficiently?
• RQ2 What is the bit complexity of computing a rational point in such a set, given bounds on degrees and coefficient sizes?
• RQ3 Can the existence of a rational sum-of-squares decomposition for a rational polynomial be decided algorithmically?
• RQ4 What is the bit complexity of computing the rational SOS representation when it exists?
• RQ5 Under what conditions does the existence of a real SOS decomposition imply the existence of a rational SOS decomposition?

Key findings

• The algorithm decides rational point existence in a convex semi-algebraic set in $\sigma^{O(1)}D^{O(k^3)}$ bit operations, where $\sigma$ bounds coefficient bit lengths and $D$ the maximum polynomial degree.
• When a rational point exists, its coordinates have bit length at most $\sigma D^{O(k^3)}$.
• The algorithm can decide whether a rational polynomial $f$ of degree $2d$ is a sum of squares in $\mathbb{Q}[x_1,\ldots,x_n]$ in $\tau^{O(1)}D^{O(k^3)}$ bit operations, where $\tau$ bounds the bit length of $f$'s coefficients.
• If $f$ is a rational SOS, the coefficients of the rational polynomials in the decomposition have bit length at most $\tau D^{O(k^3)}$.
• The method achieves a significant improvement over prior approaches: the complexity is $\tau^{O(1)}D^{O(k)}$ for the RationalTotalRealSolve function, under the condition that the solution variety has only real roots.
• The paper establishes that if a polynomial has a real SOS decomposition and the solution variety is real-rooted, then a rational SOS decomposition exists and can be computed efficiently.