A multivariate polynomial is stable if it is nonvanishing whenever all variables have positive imaginary parts. We classify all linear partial differential operators in the Weyl algebra An that preserve stability. An important tool that we develop in the process is the higher dimensional generalization of Pólya-Schur’s notion of multiplier sequence. We characterize all multivariate multiplier s...

We consider a closed set S ⊆ Rn and a linear operator Φ: R[X1, . . . , Xn]→ R[X1, . . . , Xn] that preserves nonnegative polynomials, in the following sense: if f ≥ 0 on S, then Φ(f) ≥ 0 on S as well. We show that each such operator is given by integration with respect to a measure taking nonnegative functions as its values. This can be seen as a generalization of Haviland’s Theorem, which conc...

Let A and B be n × m matrices. The matrix B is said to be g-row majorized (respectively g-column majorized) by A, if every row (respectively column) of B, is g-majorized by the corresponding row (respectively column) of A. In this paper all kinds of g-majorization are studied on Mn,m, and the possible structure of their linear preservers will be found. Also all linear operators T : Mn,m ---> Mn...