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- W4294295344 abstract "A finite linear combination of matrix elements of a finite dimensional representation $pi$ of a Lie group $K$ is said to be a $pi$-polynomial on $K$. If $K$ is compact then any $pi$-polynomial uniquely extends to a holomorphic function on the complexification $K_C$ of $K$. For a system of $n$ $pi_i$-polynomials, where $n=dim(K)$, we consider the proportion of real roots, that is the ratio of roots number to the number of roots in $K_C$. It turns out that for growing representations $pi_i$ and random system of $pi_i$-polynomials, the expected proportion of real roots converges not to $0$, but to a nonzero constant. The limit is calculated in terms of the volumes of some compact convex sets that determine the growth of representation $pi$. For a $1$-dimensional torus $K$ the limit is $1/sqrt 3$." @default.
- W4294295344 created "2022-09-02" @default.
- W4294295344 creator A5018647815 @default.
- W4294295344 date "2022-08-31" @default.
- W4294295344 modified "2023-09-25" @default.
- W4294295344 title "How many roots of a random polynomial system on a Lie group are real?" @default.
- W4294295344 doi "https://doi.org/10.48550/arxiv.2208.14711" @default.
- W4294295344 hasPublicationYear "2022" @default.
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