Rational certificates of non-negativity on semialgebraic subsets of cylinders

JERONIMO, GABRIELA; PERRUCCI, DANIEL

JOURNAL OF PURE AND APPLIED ALGEBRA

ELSEVIER SCIENCE BV

Año: 2024 vol. 228

0022-4049

Let g_1,...g_s in R[X_1,...,X_n,Y] and S= {(x,y) in R^{n+1} | g_1(x,y) ge 0, ..., g_s(x,y)ge 0} be a non-empty, possibly unbounded, subset of a cylinder in R^{n+1}. Let f in R[X_1,...,X_n,Y] be a polynomial which is positive on S. We prove that, under certain additional assumptions, for any non-constant polynomial q in R[Y] which is positive on R, there is a certificate of the non-negativity of f on S given by a rational function having as numerator a polynomial in the quadratic module generated by g_1,..., g_s and as denominator a power of q.