双圆盘上无零的多项式
Polynomials with no zeros on the bidisk
摘要：
本文证明了双圆盘$\mathbb{D}^2$上二元无零点多项式的平方和公式，推广了Cole-Wermer和Geronimo-Woerdeman的公式.该公式与单位圆上正交多项式的Christoffel-Darboux公式有关，但推广到两个变量涉及公式的唯一性问题和研究两个变量正交多项式关于可能具有无穷大质量的环面上的正Borel测度的理想。我们目前的应用程序，两个变量Fejér-Riesz分解，解析扩展定理一类的有边曲线称为杰出的品种，并挑选内插的bidisk。
英文摘要：
We prove a detailed sums of squares formula for two variable polynomials with no zeros on the bidisk $\mathbb{D}^2$ extending previous versions of such a formula due to Cole-Wermer and Geronimo-Woerdeman. The formula is related to the Christoffel-Darboux formula for orthogonal polynomials on the unit circle, but the extension to two variables involves issues of uniqueness in the formula and the study of ideals of two variable orthogonal polynomials with respect to a positive Borel measure on the torus which may have infinite mass. We present applications to two variable Fejér-Riesz factorizations, analytic extension theorems for a class of bordered curves called distinguished varieties, and Pick interpolation on the bidisk.
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