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：一个理想元素的余因子表示，即用生成元表示的表示，可以被视为理想成员关系的一个证书。这种表示通常不唯一，有些表示可能比其他表示复杂得多。在本文中，我们考虑计算多项式理想中给定元素的最稀疏余因子表示（即项数最少的表示）的问题。虽然我们专注于更一般的非交换多项式情形，但所有结果也适用于交换情形。我们证明，计算有界项数的余因子表示问题是可判定的且属于NP完全问题。此外，我们通过将该问题转化为线性优化问题，并利用基于签名的Gröbner基算法的性质，提出了一种计算稀疏（不一定最优）表示的实用算法。我们证明，对于某类理想，该方法计算出的表示实际上是最优的，并给出了实验数据，表明该方法能显著产生更稀疏的余因子表示。