论文标题
半代数的半代数描述在多项式下的半代数集
Semi-algebraic description of the closure of the image of a semi-algebraic set under a polynomial
论文作者
论文摘要
给定一个多项式$ f $和一个半代数套件$ s $，我们提供了一种符号算法来查找定义半代数集$ q $的方程式和不等式，该方程与$ s $ of $ s $ of $ s $相同，$ s $ of $ s $ of $ s $ of $ s $ f $ of $ f $ s of $ f $ s of $ s $ f $ s of f $ s $ f $，即\ ebent {等式{等式{等式{equate {equate {equate {equate {equate {equate {equate {equate {equate {q = c = cy} \end{equation} Consequently, every polynomial optimization problem whose optimum value is finite has an equivalent form with attained optimum value, i.e., \begin{equation} \min \limits_{t\in Q} t =\inf\limits_{x\in S} f(x) \end{equation} whenever the right-hand side is finite.给定$ d $作为$ f $的上限和定义$ s $的多项式的上限，我们证明我们的方法需要$ o（d^{o（n）}）$算术操作，以在最多$ d^{o（n）} $定义$ \ \ \\ overline $ \ overline {f（s s f（s s s）} $的最多生产$ d^{o（n）} $。
Given a polynomial $f$ and a semi-algebraic set $S$, we provide a symbolic algorithm to find the equations and inequalities defining a semi-algebraic set $Q$ which is identical to the closure of the image of $S$ under $f$, i.e., \begin{equation} Q=\overline{f(S)}\,. \end{equation} Consequently, every polynomial optimization problem whose optimum value is finite has an equivalent form with attained optimum value, i.e., \begin{equation} \min \limits_{t\in Q} t =\inf\limits_{x\in S} f(x) \end{equation} whenever the right-hand side is finite. Given $d$ as the upper bound on the degrees of $f$ and polynomials defining $S$, we prove that our method requires $O(d^{O(n)})$ arithmetic operations to produce polynomials of degrees at most $d^{O(n)}$ defining $\overline{f(S)}$.
