A cofactor representation of an ideal element, that is, a representation in terms of the generators, can be considered as a certificate for ideal membership. Such a representation is typically not unique, and some can be a lot more complicated than others. In this work, we consider the problem of computing sparsest cofactor representations, i.e., representations with a minimal number of terms, of a given element in a polynomial ideal. While we focus on the more general case of noncommutative polynomials, all results also apply to the commutative setting. We show that the problem of computing cofactor representations with a bounded number of terms is decidable and NP-complete. Moreover, we provide a practical algorithm for computing sparse (not necessarily optimal) representations by translating the problem into a linear optimization problem and by exploiting properties of signature-based Gr\"obner basis algorithms. We show that for a certain class of ideals, representations computed by this method are actually optimal, and we present experimental data illustrating that it can lead to noticeably sparser cofactor representations.

翻译：理想元素的余因子表示，即用生成元表示的形式，可视为理想成员关系的证书。这类表示通常不唯一，且某些表示可能远比其它表示复杂。本文考虑计算多项式理想中给定元素的最稀疏余因子表示问题，即具有最少项数的表示。尽管我们主要关注非交换多项式这一更一般情形，但所有结论同样适用于交换情形。我们证明了有界项数余因子表示问题的可判定性及其NP完全性。此外，通过将该问题转化为线性优化问题并利用基于签名的Gröbner基算法性质，我们提出了一种实用算法来计算稀疏（不必最优）表示。我们证明，对于特定类别的理想，该方法计算出的表示实际上是最优的，并给出实验数据表明该方法能生成明显更稀疏的余因子表示。