Geodesic
Schur polynomials and matrix positivity preservers
Discrete mathematics & theoretical computer science, DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016), DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016) (2020) Cet article a éte moissonné depuis la source Episciences

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A classical result by Schoenberg (1942) identifies all real-valued functions that preserve positive semidefi- niteness (psd) when applied entrywise to matrices of arbitrary dimension. Schoenberg's work has continued to attract significant interest, including renewed recent attention due to applications in high-dimensional statistics. However, despite a great deal of effort in the area, an effective characterization of entrywise functions preserving positivity in a fixed dimension remains elusive to date. As a first step, we characterize new classes of polynomials preserving pos- itivity in fixed dimension. The proof of our main result is representation theoretic, and employs Schur polynomials. An alternate, variational approach also leads to several interesting consequences including (a) a hitherto unexplored Schubert cell-type stratification of the cone of psd matrices, (b) new connections between generalized Rayleigh quo- tients of Hadamard powers and Schur polynomials, and (c) a description of the joint kernels of Hadamard powers. 
@article{DMTCS_2020_special_379_a90,
     author = {Belton, Alexander and Guillot, Dominique and Khare, Apoorva and Putinar, Mihai},
     title = {Schur polynomials and matrix positivity preservers},
     journal = {Discrete mathematics & theoretical computer science},
     year = {2020},
     volume = {DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016)},
     doi = {10.46298/dmtcs.6408},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.6408/}
}
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%D 2020
%V DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016)
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Belton, Alexander; Guillot, Dominique; Khare, Apoorva; Putinar, Mihai. Schur polynomials and matrix positivity preservers. Discrete mathematics & theoretical computer science, DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016), DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016) (2020). doi: 10.46298/dmtcs.6408

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