SIAM Review
Certifying inconsistency of sparse linear systems
ISSAC '98 Proceedings of the 1998 international symposium on Symbolic and algebraic computation
Global Optimization with Polynomials and the Problem of Moments
SIAM Journal on Optimization
Computing sum of squares decompositions with rational coefficients
Theoretical Computer Science
Proceedings of the 2009 conference on Symbolic numeric computation
Verifying nonlinear real formulas via sums of squares
TPHOLs'07 Proceedings of the 20th international conference on Theorem proving in higher order logics
Proceedings of the 2010 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
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We deploy numerical semidefinite programming and conversion to exact rational inequalities to certify that for a positive semidefinite input polynomial or rational function, any representation as a fraction of sums-of-squares of polynomials with real coefficients must contain polynomials in the denominator of degree no less than a given input lower bound. By Artin's solution to Hilbert's 17th problems, such representations always exist for some denominator degree. Our certificates of infeasibility are based on the generalization of Farkas's Lemma to semidefinite programming. The literature has many famous examples of impossibility of SOS representability including Motzkin's, Robinson's, Choi's and Lam's polynomials, and Reznick's lower degree bounds on uniform denominators, e.g., powers of the sum-of-squares of each variable. Our work on exact certificates for positive semidefiniteness allows for non-uniform denominators, which can have lower degree and are often easier to convert to exact identities. Here we demonstrate our algorithm by computing certificates of impossibilities for an arbitrary sum-of-squares denominator of degree 2 and 4 for some symmetric sextics in 4 and 5 variables, respectively. We can also certify impossibility of base polynomials in the denominator of restricted term structure, for instance as in Landau's reduction by one less variable.