Given a polynomial ring P over a field K, an element $g\in P$, and a K-subalgebra S of P, we deal with the problem of saturating S with respect to g, i.e. computing ${\mathrm{Sat}}_g(S)=S[g,g^{-1}]\cap P$. In the general case we describe a procedure/algorithm to compute a set of generators for ${\mathrm{Sat}}_g(S)$ which terminates if and only if it is finitely generated. Then we consider the more interesting case when S is graded. In particular, if S is graded by a positive matrix W and g is an indeterminate, we show that if we choose a term ordering σ of g-DegRev type compatible with W, then the two operations of computing a σ-SAGBI basis of S and saturating S with respect to g commute. This fact opens the doors to nice algorithms for the computation of ${\mathrm{Sat}}_g(S)$. In particular, under special assumptions on the grading one can use the truncation of a σ-SAGBI basis and get the desired result. Notably, this technique can be applied to the problem of directly computing some U-invariants, classically called semi-invariants, even in the case that K is not the field of complex numbers.

给定域K 上的多项式环P，一个元素$G\in 磷$和ķ -subalgebra š的P，我们处理饱和的问题小号相对于克，即计算${\mathrm{星期六}}_G(秒)=秒[G,G^{-1}]\cap 磷$. 在一般情况下，我们描述了一个过程/算法来计算一组生成器${\mathrm{星期六}}_G(秒)$终止当且仅当它是有限生成的。然后我们考虑更有趣的情况，当S被分级时。特别地，如果S由正矩阵W分级并且g是不确定的，我们表明如果我们选择与W兼容的g - DegRev类型的项排序σ，那么计算S的σ -SAGBI 基的两个操作并且关于g通勤使S饱和。这个事实打开了用于计算的好算法的大门${\mathrm{星期六}}_G(秒)$. 特别是，在分级的特殊假设下，可以使用σ -SAGBI 基的截断并获得所需的结果。值得注意的是，即使在K不是复数域的情况下，这种技术也可以应用于直接计算一些U不变量的问题，经典上称为半不变量。