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- W4320342337 abstract "Investigating a problem posed by W. Hengartner (2000), we study the maximal valence (number of preimages of a prescribed point in the complex plane) of logharmonic polynomials, i.e., complex functions that take the form $f(z) = p(z) overline{q(z)}$ of a product of an analytic polynomial $p(z)$ of degree $n$ and the complex conjugate of another analytic polynomial $q(z)$ of degree $m$. In the case $m=1$, we adapt an indirect technique utilizing anti-holomorphic dynamics to show that the valence is at most $3n-1$. This confirms a conjecture of Bshouty and Hengartner (2000). Using a purely algebraic method based on Sylvester resultants, we also prove a general upper bound for the valence showing that for each $n,m geq 1$ the valence is at most $n^2+m^2$. This improves, for every choice of $n,m geq 1$, the previously established upper bound $(n+m)^2$ based on Bezout's theorem. We also consider the more general setting of polyanalytic polynomials where we show that this latter result can be extended under a nondegeneracy assumption." @default.
- W4320342337 created "2023-02-13" @default.
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- W4320342337 date "2023-02-08" @default.
- W4320342337 modified "2023-09-23" @default.
- W4320342337 title "On the valence of logharmonic polynomials" @default.
- W4320342337 doi "https://doi.org/10.48550/arxiv.2302.04339" @default.
- W4320342337 hasPublicationYear "2023" @default.
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