论文标题
关于lojasiewicz的不平等现象和有效的普替纳尔人的Potitivstellensatz
On Łojasiewicz Inequalities and the Effective Putinar's Positivstellensatz
论文作者
论文摘要
在正方形之和的半代数集上，正多项式在一个真实代数几何形状中是一个核心问题，potitivstellensatz回答了正方形。在本文中，我们在紧凑的基本半代数集$ s $上研究了有效的普通人的Potitivestellensatz，并在积极多项式的代表程度上提供了新的证明和新的改进界限。这些新界限涉及一个参数$ε$，测量正函数的非变化，常数$ \ mathfrak {c} $和ugjasiewicz的指数$ l $ a lojasiewicz不平等的不平等现象与不平等相关的半代数距离函数$ \ mathbf {g_1 = $ s $ s $ s $ s $ s $ s $ s，它们在$ \ Mathfrak {c} $和$ε^{ - 1} $中是多项式，仅根据$ L $而具有指数。当定义不平等$ \ mathbf g $满足约束资格条件时，我们详细分析了lojasiewicz的不平等现象。我们表明，在这种情况下，lojasiewicz指数$ l $是$ 1 $，我们将lojasiewicz常数$ \ mathfrak {c} $与$ \ mathbf g $的距离与单个单数系统集相关联。
The representation of positive polynomials on a semi-algebraic set in terms of sums of squares is a central question in real algebraic geometry, which the Positivstellensatz answers. In this paper, we study the effective Putinar's Positivestellensatz on a compact basic semi-algebraic set $S$ and provide a new proof and new improved bounds on the degree of the representation of positive polynomials. These new bounds involve a parameter $ε$ measuring the non-vanishing of the positive function, the constant $\mathfrak{c}$ and exponent $L$ of a Łojasiewicz inequality for the semi-algebraic distance function associated to the inequalities $\mathbf{g} = (g_1, \dots , g_r)$ defining $S$. They are polynomial in $\mathfrak{c}$ and $ε^{-1}$ with an exponent depending only on $L$. We analyse in details the Łojasiewicz inequality when the defining inequalities $\mathbf g$ satisfy the Constraint Qualification Condition. We show that, in this case, the Łojasiewicz exponent $L$ is $1$ and we relate the Łojasiewicz constant $\mathfrak{c}$ with the distance of $\mathbf g$ to the set of singular systems.
