Let @math be a linear matrix, or pencil, generated by given symmetric matrices @math of size @math with rational entries. The set of real vectors x such that the pencil is positive semidefinite is a convex semi-algebraic set called spectrahedron, described by a linear matrix inequality (LMI). We design an exact algorithm that, up to genericity assumptions on the input matrices, computes an exact algebraic representation of at least one point in the spectrahedron, or decides that it is empty. The algorithm does not assume the existence of an interior point, and the computed point minimizes the rank of the pencil on the spectrahedron. The degree @math of the algebraic representation of the point coincides experimentally with the algebraic degree of a generic semidefinite program associated to the pencil. We provide explicit bounds for the complexity of our algorithm, proving that the maximum number of arithmetic operations that are performed is essentially quadratic in a multilinear B 'ezout bound of @math . When @math (resp. @math ) is fixed, such a bound, and hence the complexity, is polynomial in @math (resp. @math ). We conclude by providing results of experiments showing practical improvements with respect to state-of-the-art computer algebra algorithms.
Also, in @cite_49 , a version of @cite_40 dedicated to spectrahedra exploiting some of their structural properties, decides whether a linear matrix inequality @math has a rational solution, that is whether @math contains a point with coordinates in @math . Remark that such an algorithm is not sufficient to solve our problem, since, in some degenerate but interesting cases, @math is not empty but does not contain rational points: in Section we will illustrate the application of our algorithm to one of these examples.
@cite_40 @cite_49
2064131066 2011936157
Let @math , @math for @math , @math bounding the bit length of the coefficients of the @math 's, and let @math be a quantifier-free @math -formula defining a convex semialgebraic set. We design an algorithm returning a rational point in @math if and only if @math . It requires @math bit operations. If a rational point is outputted, its coordinates have bit length dominated by @math . Using this result, we obtain a procedure for deciding whether a polynomial @math is a sum of squares of polynomials in @math . Denote by @math the degree of @math , @math the maximum bit length of the coefficients in @math , @math , and @math . This procedure requires @math bit operations, and the coefficients of the outputted polynomials have bit length dominated by @math . 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( (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.