Vincent 定理指出: 若~\(f(x)\) 为~\(d\) 次实系数多项式, $(a_1,b_1)$ 为开区间, 则多项式~\(f(x)\) 在~\((a_1,\ b_1)\) 上没有实根当且仅当存在正常数 $\delta$, 使得对任意区间 $(a,b)\subset (a_1,b_1)$ , 当 $|a-b|<\delta$ 时, 多项式 ~\((1+x)^d f(\frac{a+bx}{1+x})\) 的系数不变号\ (都是正数或都是负数). 文章的主要工作是推广这一结果到一般的多变元代数系统. 设实系数多项式 ~\(f\in \mathbb{R}[x_1,x_2, \cdots, x_n]\), $f$ 相对于变元 $x_i$ 的次数记为 $d_i$. 记区间的笛卡尔积为 ~ $I=[a_1,b_1]\times [a_2,b_2]\times \cdots \times [a_n,b_n]$ (也称为Box). 记~\(\phi(I)=\max\{b_i-a_i,\ i=1,2,\cdots,n\}\). 定义 $$f_I=(1+x_1{)}^{d_1}(1+x_2{)}^{d_2}\cdots (1+x_n{)}^{d_n}f(\frac{a_1+b_1{x}_1}{1+x_1},\frac{a_2+b_2{x}_2}{1+x_2},\cdots ,\frac{a_n+b_n{x}_n}{1+x_n}).$$ 称 $f_I$ 为 $f$ 相对于\ Box $I$ 的伴随多项式. 证明了: 若多项式~\(f_1,f_2,\cdots, f_m \in \mathbb{R}[x_1,x_2,\cdots, x_n], \) 且Box $\text{It}\mathrm{\Lambda }\subset {\mathbb{R}}^n$, 则方程组 $\{f_1=0,f_2=0,\cdots ,f_m=0$\} 在Box $\text{It}\mathrm{\Lambda }$ 上没有零点, 当且仅当存在正常数 $\delta$ (与Box $\text{It}\mathrm{\Lambda }$ 有关), 使得 对于任意Box $I\subset \text{It}\mathrm{\Lambda }$, 当 $\varphi (I)<\delta$ 时, 伴随多项式 $$f_{1I},f_{2I},\cdots ,f_{mI}$$ 中至少一个 $f_{iI}$ 的非零系数全是正\ (或负) 数且 $f_i$ 在Box $I$ 的所有顶点上的值不为~0.

Let $f(x)$ be a real polynomial of degree $d$ and $(a_1,b_1)$ be an open interval. Vincent's Theorem indicates that there is a positive quantity $\delta$ so that for every pair of real numbers $a,b((a,b)\subset (a_1,b_1))$ with $|b-a|<\delta$, the polynomial of the form $f_{(a,b)}=(1+x{)}^{d}f(\frac{a+bx}{1+x})$ has exactly 0 variations (non-zero coefficients of $f_{(a,b)}$ are all positive or all negative) if and only if $f(x)$ has no real root over the interval $(a_1,b_1)$. In this paper, we generalize this result to system of multivariate polynomials. Let ~\(f\in \mathbb{R}[x_1,x_2,\cdots, x_n]\), where the degree of $x_i$ is $d_i$. Let~ $I=[a_1,b_1]\times [a_2,b_2]\times \cdots \times [a_n,b_n]$ be a Cartesian product of $n$ intervals (also called Box), and let ~\(\phi(I)=\max\{b_i-a_i,\ i=1,2,\cdots,n\}\). Define $$f_I=(1+x_1{)}^{d_1}(1+x_2{)}^{d_2}\cdots (1+x_n{)}^{d_n}f(\frac{a_1+b_1{x}_1}{1+x_1},\frac{a_2+b_2{x}_2}{1+x_2},\cdots ,\frac{a_n+b_n{x}_n}{1+x_n}).$$ Here $f_I$ is called an adjoint polynomial of $f$ relating to Box $I$. In this paper, we proved the following result. Let \(f_1,f_2,\cdots, f_m \in \mathbb{R}[x_1,x_2,\cdots, x_n], \) and Box $\text{It}\mathrm{\Lambda }\subset {\mathbb{R}}^n$. Then the the system of equations $\{f_1=0,f_2=0,\cdots ,f_m=0$\} has no zero on the Box $\text{It}\mathrm{\Lambda }$, if and only if there is a positive quantity $\delta$ (relating to Box $\text{It}\mathrm{\Lambda }$), such that: (i) For any Box $I\subset \text{It}\mathrm{\Lambda }$, when $\varphi (I)<\delta$, at least one adjoint polynomial $f_{iI}$ has exactly 0 variations in $f_{1I},f_{2I},\cdots ,f_{mI}$; (ii) The value of $f_i$ corresponding to $f_{iI}$ is not 0 at all vertices of Box $I$.