自由Bertini定理及其应用

摘要：

贝尔蒂尼不可约定理的最简单版本指出，非复合多项式函数的一般纤维是不可约超曲面。本文的主要结果是它对自由代数的类比：如果$f$是一个非交换多项式，使得$f-λ$对无穷多个标量$λ$有因子，则存在一个非交换多项式$h$和一个非常数一元多项式$p$，使得$f=p\circ h$ .给出了自由Bertini定理在非交换多项式矩阵求值中的两个应用。$f$的特征值集是所有矩阵元组$X$的集合，其中$f(X)$达到某个给定的特征值。证明了$f$和$g$的特征值集重合的充要条件是对某个非零非交换多项式$a$，满足$fa=ag$.第二个应用程序属于拟凸性和描述多项式$f$，使连接组件的$\{X \text{ tuple of symmetric $ n \times n $ matrices}: λI\succ f(X) \}$关于原点是凸的所有自然$n$和$λ>0$。它表明，这样的多项式是处处负半定或一元和凸二次多项式的组合。

英文摘要：

The simplest version of Bertini's irreducibility theorem states that the generic fiber of a non-composite polynomial function is an irreducible hypersurface. The main result of this paper is its analog for a free algebra: if $f$ is a noncommutative polynomial such that $f-λ$ factors for infinitely many scalars $λ$, then there exist a noncommutative polynomial $h$ and a nonconstant univariate polynomial $p$ such that $f=p\circ h$. Two applications of free Bertini's theorem for matrix evaluations of noncommutative polynomials are given. An eigenlevel set of $f$ is the set of all matrix tuples $X$ where $f(X)$ attains some given eigenvalue. It is shown that eigenlevel sets of $f$ and $g$ coincide if and only if $fa=ag$ for some nonzero noncommutative polynomial $a$. The second application pertains quasiconvexity and describes polynomials $f$ such that the connected component of $\{X \text{ tuple of symmetric $n\times n$ matrices}: λI\succ f(X) \}$ about the origin is convex for all natural $n$ and $λ>0$. It is shown that such a polynomial is either everywhere negative semidefinite or the composition of a univariate and a convex quadratic polynomial.

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