The computational complexity of simultaneous diophantine approximation problems
SIAM Journal on Computing
SIAM Review
Matrix computations (3rd ed.)
Real solving for positive dimensional systems
Journal of Symbolic Computation
On the complexity of Putinar's Positivstellensatz
Journal of Complexity
The voronoi diagram of three lines
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
Proceedings of the 2007 international workshop on Symbolic-numeric computation
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
Computing sum of squares decompositions with rational coefficients
Theoretical Computer Science
Global optimization of polynomials using generalized critical values and sums of squares
Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation
Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation
Deciding reachability of the infimum of a multivariate polynomial
Proceedings of the 36th international symposium on Symbolic and algebraic computation
Quadratic-time certificates in linear algebra
Proceedings of the 36th international symposium on Symbolic and algebraic computation
The minimum-rank gram matrix completion via modified fixed point continuation method
Proceedings of the 36th international symposium on Symbolic and algebraic computation
On the generation of positivstellensatz witnesses in degenerate cases
ITP'11 Proceedings of the Second international conference on Interactive theorem proving
Computing Rational Points in Convex Semialgebraic Sets and Sum of Squares Decompositions
SIAM Journal on Optimization
Proceedings of the 2011 International Workshop on Symbolic-Numeric Computation
Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation
Termination conditions for positivity proving procedures
Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation
Computing rational solutions of linear matrix inequalities
Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation
Algorithmic aspects of sums of Hermitian squares of noncommutative polynomials
Computational Optimization and Applications
Certification of bounds of non-linear functions: the templates method
CICM'13 Proceedings of the 2013 international conference on Intelligent Computer Mathematics
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We present a hybrid symbolic-numeric algorithm for certifying a polynomial or rational function with rational coefficients to be non-negative for all real values of the variables by computing a representation for it as a fraction of two polynomial sum-of-squares (SOS) with rational coefficients. Our new approach turns the earlier methods by Peyrl and Parrilo at SNC'07 and ours at ISSAC'08 both based on polynomial SOS, which do not always exist, into a universal algorithm for all inputs via Artin's theorem. Furthermore, we scrutinize the all-important process of converting the numerical SOS numerators and denominators produced by block semidefinite programming into an exact rational identity. We improve on our own Newton iteration-based high precision refinement algorithm by compressing the initial Gram matrices and by deploying rational vector recovery aside from orthogonal projection. We successfully demonstrate our algorithm on (1) various exceptional SOS problems with necessary polynomial denominators from the literature and on (2) very large (thousands of variables) SOS lower bound certificates for Rump's model problem (up to n=18, factor degree=17).