On the combinatorial and algebraic complexity of quantifier elimination
Journal of the ACM (JACM)
Computing integral points in convex semi-algebraic sets
FOCS '97 Proceedings of the 38th Annual Symposium on Foundations of Computer Science
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
Computing the global optimum of a multivariate polynomial over the reals
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
Computing sum of squares decompositions with rational coefficients
Theoretical Computer Science
Proceedings of the 2009 conference on Symbolic numeric computation
Deciding reachability of the infimum of a multivariate polynomial
Proceedings of the 36th international symposium on Symbolic and algebraic computation
Journal of Symbolic Computation
Computing Rational Points in Convex Semialgebraic Sets and Sum of Squares Decompositions
SIAM Journal on Optimization
Global optimization of polynomials restricted to a smooth variety using sums of squares
Journal of Symbolic Computation
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Consider a (D x D) symmetric matrix A whose entries are linear forms in Q[X1, ..., Xk] with coefficients of bit size ≤ τ. We provide an algorithm which decides the existence of rational solutions to the linear matrix inequality A ≥ 0 and outputs such a rational solution if it exists. This problem is of first importance: it can be used to compute algebraic certificates of positivity for multivariate polynomials. Our algorithm runs within (k≤)O(1)2O(min(k, D)D2)DO(D2) bit operations; the bit size of the output solution is dominated by τO(1)2O(\min(k, D)D2). These results are obtained by designing algorithmic variants of constructions introduced by Klep and Schweighofer. This leads to the best complexity bounds for deciding the existence of sums of squares with rational coefficients of a given polynomial. We have implemented the algorithm; it has been able to tackle Scheiderer's example of a multivariate polynomial that is a sum of squares over the reals but not over the rationals; providing the first computer validation of this counter-example to Sturmfels' conjecture.