Assuming sufficiently many terms of a n-dimensional table defined over a field are given, we aim at guessing the linear recurrence relations with either constant or polynomial coefficients they satisfy. In many applications, the table terms come along with a structure: for instance, they may be zero outside of a cone, they may be built from a Gr{\"o}bner basis of an ideal invariant under the action of a finite group. Thus, we show how to take advantage of this structure to both reduce the number of table queries and the number of operations in the base field to recover the ideal of relations of the table. In applications like in combinatorics, where all these zero terms make us guess many fake relations, this allows us to drastically reduce these wrong guesses. These algorithms have been implemented and, experimentally, they let us handle examples that we could not manage otherwise. Furthermore, we show which kind of cone and lattice structures are preserved by skew polynomial multiplication. This allows us to speed up the guessing of linear recurrence relations with polynomial coefficients by computing sparse Gr{\"o}bner bases or Gr{\"o}bner bases of an ideal invariant under the action of a finite group in a ring of skew polynomials.

中文翻译：

---

猜测序列关系的结构理想的 Gr{\"o}bner 基

假设给定了在一个域上定义的 n 维表的足够多的项，我们的目标是用它们满足的常数或多项式系数来猜测线性递推关系。在许多应用中，表项都带有一个结构：例如，它们在锥外可能为零，它们可能是由有限群作用下的理想不变量的 Gr{\"o}bner 基构建的. 因此，我们展示了如何利用这种结构来减少表查询的数量和基字段中的操作数量来恢复表的理想关系。在像组合学这样的应用中，所有这些零项让我们猜测许多虚假的关系，这使我们能够大大减少这些错误的猜测。这些算法已经实现，并且在实验中，它们让我们可以处理我们无法通过其他方式处理的示例。此外，我们展示了通过偏斜多项式乘法保留了哪种锥体和晶格结构。这使我们能够通过计算一个理想不变量的稀疏 Gr{\"o}bner 基或 Gr{\"o}bner 基在环中的有限群的作用下，加速对多项式系数的线性递推关系的猜测。偏斜多项式。