论文标题
关于多项式函数的连续选择
On continuous selections of polynomial functions
论文作者
论文摘要
连续选择多项式函数是一个连续的函数，可以将其域分区分为有限的许多片段，该函数与多项式一致。考虑到一组有限的多项式，我们表明它只有有限的连续选择，并且每个选择都是半代数的。然后，我们建立了有关这些连续选择的Clarke细分定义的关键点的一些通用属性。特别是，给定一组具有通用系数的多项式有限的多项式，我们表明，所有连续选择的临界点都是有限的，临界值都是不同的，我们还得出了从下面界定的连续选择的强制性。我们指出，有关lojasiewicz的不等式和错误界限的一些现有结果，对于某些有限的多项式的最大函数也适用于所有连续选择。
A continuous selection of polynomial functions is a continuous function whose domain can be partitioned into finitely many pieces on which the function coincides with a polynomial. Given a set of finitely many polynomials, we show that there are only finitely many continuous selections of it and each one is semi-algebraic. Then, we establish some generic properties regarding the critical points, defined by the Clarke subdifferential, of these continuous selections. In particular, given a set of finitely many polynomials with generic coefficients, we show that the critical points of all continuous selections of it are finite and the critical values are all different, and we also derive the coercivity of those continuous selections which are bounded from below. We point out that some existing results about Łojasiewicz's inequality and error bounds for the maximum function of some finitely many polynomials are also valid for all the continuous selections of them.
