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：具有奇异性的函数众所周知难以用传统逼近方案进行近似。在计算应用中，通常采用低阶分段多项式、多级方案或其他类型的分级策略进行处理。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，其在有理次数上具有根指数收敛性。这通常通过极点向奇异性附近聚集实现。函数逼近中有理函数的理论和计算实践均集中于单变量情形，并通过与复平面等同的方式扩展至二维。相比之下，多变量有理函数（即多变量多项式的商）的探索相对不足。然而，除了理论复杂度急剧增加外，它们也提供了丰富的可能性。首先观察到，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到曲线，我们探索了具有曲线奇异性函数的多变量有理逼近的构造方法。