Functions with singularities are notoriously difficult to approximate with conventional approximation schemes. In computational applications they are often resolved with low-order piecewise polynomials, multilevel schemes or other types of grading strategies. Rational functions are an exception to this rule: for univariate functions with point singularities, such as branch points, rational approximations exist with root-exponential convergence in the rational degree. This is typically enabled by the clustering of poles near the singularity. Both the theory and computational practice of rational functions for function approximation have focused on the univariate case, with extensions to two dimensions via identification with the complex plane. Multivariate rational functions, i.e., quotients of polynomials of several variables, are relatively unexplored in comparison. Yet, apart from a steep increase in theoretical complexity, they also offer a wealth of opportunities. A first observation is that singularities of multivariate rational functions may be continuous curves of poles, rather than isolated ones. By generalizing the clustering of poles from points to curves, we explore constructions of multivariate rational approximations to functions with curves of singularities.

翻译：具有奇异性的函数在使用传统逼近方案时通常难以处理。在计算应用中，这类函数常通过低阶分片多项式、多层级方案或其他类型的分层策略来求解。有理函数是这一规则的例外：对于具有点奇异性（如分支点）的单变量函数，存在有理逼近，其在有理度下具有根指数收敛性。这通常通过极点向奇异点附近的聚集来实现。有理函数在函数逼近领域的理论与计算实践主要集中于单变量情形，并通过与复平面的对应扩展到二维。相比之下，多变量有理函数（即多个变量多项式之商）的研究相对较少。然而，除了理论复杂性的显著增加外，这类函数也提供了丰富的研究机遇。首要发现是，多变量有理函数的奇异性可能是连续的极点曲线，而非孤立点。通过将极点聚集从点推广至曲线，我们探索了针对具有奇异性曲线的函数构造多变量有理逼近的方法。