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Session S24 - Symbolic Computation: Theory, Algorithms and Applications

Tuesday, July 20, 18:00 ~ 18:25 UTC-3

Univariate Rational Sum of Squares

Agnes Szanto

North Carolina State University, United States   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

It is well-known that a non-negative univariate real polynomial is a sum of 2 squares of real polynomials. Landau in 1905 proved that every univariate polynomial with rational coefficients which is strictly positive on R is a sum of 8 squares of rational polynomials. However, for rational polynomials that are non-negative on R, Scheiderer in 2013 constructed examples which are not sums of squares of rational polynomials. In this talk we consider the local case, namely, polynomials that are non-negative on the real roots of another non-zero polynomial. Parrilo in 2003 gave a simple construction that implies that if f in R[x] is square-free and g in R[x] is non-negative on the real roots of f then g is a sum of squares of real polynomials modulo f. We extend this result to the case when f is an arbitrary rational polynomial and g in Q[x] is non-negative on the real roots of f, assuming that gcd(f/d,d)=1 for d=gcd(f,g). In this case we prove that g is a positive rational combination of squares of rational polynomials modulo f. Moreover, we give bounds on the size of such rational sum of square decomposition and compare algorithmic approaches to compute one.

Joint work with Teresa Krick (Universidad de Buenos Aires) and Bernard Mourrain (INRIA, Sophia Antipolis).

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