Orthogonal polynomials in two variables on cubic curves are considered, including the case of elliptic curves. For an integral with respect to an appropriate weight function defined on a cubic curve, an explicit basis of orthogonal polynomials is constructed in terms of two families of orthogonal polynomials in one variable. We show that these orthogonal polynomials can be used to approximate functions with cubic and square root singularities, and demonstrate their usage for solving differential equations with singular solutions.

中文翻译：

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平面三次曲线上的正交多项式

考虑了三次曲线上两个变量的正交多项式，包括椭圆曲线的情况。对于与在三次曲线上定义的适当权函数有关的积分，在一个变量中根据正交多项式的两个族构造了正交多项式的显式基础。我们证明了这些正交多项式可用于近似具有三次和平方根奇异性的函数，并证明了它们在求解具有奇异解的微分方程中的用法。