Sergey Sverchkov | Searcharxiv

arXiv2026

A Counterexample to the Global Injectivity of a Jacobian Mapping in \mathbb{C}^5 and Its Analytical Roots

We study a specific polynomial mapping G: \mathbb{C}^5 \to \mathbb{C}^5 induced by a homogeneous polynomial F of degree 6 consisting of four tridiagonal harmonic blocks. We prove that while the Jacobian matrix of this mapping is unipotent at every point (implying det J_G(x) \equiv 1), the mapping itself is not globally injective. We construct explicit algebraic sparse pairs of distinct points a \neq b that map to the identical image G(a) = G(b) = \vec{0}. Furthermore, we perform a comprehensive classification of the zero-set structure of the corresponding gradient field, uncovering a total of 37 precise analytical complex solutions split across six distinct geometric series.

math.AC↗

arXiv2022

New elements in the center of free alternative algebra

A new series of central elements is found in the free alternative algebra. More exactly, let $Alt[X]$ and $SMalc[X]\subset Alt[X]$ be the free alternative algebra and the free special Malcev algebra over a field of characteristic 0 on a set of free generators $X$, and let $f(x,y,x_1,\ldots,x_n)\in SMalc[X]$ be a multilinear element which is trivial in the free associative algebra. Then the element $u_n=u_n(x,x_1,\ldots,x_n)=f(x^2,x,x_1,\ldots,x_n)-f(x,x^2,x_1,\ldots,x_n)$ lies in the center of the algebra $Alt[X]$. The elements $u_n(x,x_1,\ldots,x_n)$ are uniquely defined up to a scalar for a given $n$, and they are skew-symmetric on the variables $x_1,\ldots,x_n$. Moreover, $u_n=0$ for $n=4m+2,\,4m+3$. and $u_n\neq 0$ for $n=4m,4m+1$. The ideals generated by the elements $u_{4m},\,u_{4m+1}$ lie in the associative center of the algebra $Alt[X]$ and have trivial multiplication.

math.RA↗