论文标题
佩尔的方程，平方英尺的平衡度量和平衡度量
Pell's equation, sum-of-squares and equilibrium measures of a compact set
论文作者
论文摘要
我们首先将pell的方程式解释为每个度t的Chebyshev多项式所满足，作为一定的activstellensatz，然后为每个整数t屈服，我们称之为广义的pell方程，由“'级''2t的基督佛尔函数相关的'2t，与''''2t的互惠所满足，与等价级别的$ unterme $ $ $ unter-$ $ $ $ 2，1]和1 $ 1，1]，1] $ 1）。接下来，我们将此观点扩展到任意紧凑的基本半代数集s $ \ subset $ r n并获得广义的pell方程（通过类似于间隔[-1，1]）。在某些条件下，对于每个t，将方程式通过基督佛尔的函数的倒数来满足“学位” 2T 2T与（i）相关的（i）平衡度量的$ $ $ $ $ $ $ $的s和（ii），对于适当的发电机g的gd $μ$。这些方程的适当发电机g。这些方程取决于定义设置的特定发电机选择。另外还显示了3个= 1个Intel Intel Intel Intel Intel Intel Intel Inteval [-1]，我们是1]。还满足了2d-simplex，2d-Euclidean单位球和单元盒的平衡度量。有趣的是，该观点在一侧连接正交多项式，基督教派函数和平衡度量，并在另一侧具有实体总和优化和阳性证书的总和。
We first interpret Pell's equation satisfied by Chebyshev polynomials for each degree t, as a certain Positivstellensatz, which then yields for each integer t, what we call a generalized Pell's equation, satisfied by reciprocals of Christoffel functions of ''degree'' 2t, associated with the equilibrium measure $μ$ of the interval [--1, 1] and the measure (1 -- x 2)d$μ$. We next extend this point of view to arbitrary compact basic semi-algebraic set S $\subset$ R n and obtain a generalized Pell's equation (by analogy with the interval [--1, 1]). Under some conditions, for each t the equation is satisfied by reciprocals of Christoffel functions of ''degree'' 2t associated with (i) the equilibrium measure $μ$ of S and (ii), measures gd$μ$ for an appropriate set of generators g of S. These equations depend on the particular choice of generators that define the set S. In addition to the interval [--1, 1], we show that for t = 1, 2, 3, the equations are indeed also satisfied for the equilibrium measures of the 2D-simplex, the 2D-Euclidean unit ball and unit box. Interestingly, this view point connects orthogonal polynomials, Christoffel functions and equilibrium measures on one side, with sum-of-squares, convex optimization and certificates of positivity in real algebraic geometry on another side.
