Can an ideal I in a polynomial ring \(\Bbbk [\mathbf {x}]\) over a field be moved by a change of coordinates into a position where it is generated by binomials \(\mathbf {x}^\mathbb A- \lambda \mathbf {x}^\mathbf {b}\) with \(\lambda \in \Bbbk \), or by unital binomials (i.e., with \(\lambda = 0\) or 1)? Can a variety be moved into a position where it is toric? By fibering the G-translates of I over an algebraic group G acting on affine space, these problems are special cases of questions about a family \(\mathcal {I}\) of ideals over an arbitrary base B. The main results in this general setting are algorithms to find the locus of points in B over which the fiber of \(\mathcal {I}\)

• is contained in the fiber of a second family \(\mathcal {I}'\) of ideals over B;
• defines a variety of dimension at least d;
• is generated by binomials; or
• is generated by unital binomials.

A faster containment algorithm is also presented when the fibers of \(\mathcal {I}\) are prime. The big-fiber algorithm is probabilistic but likely faster than known deterministic ones. Applications include the setting where a second group T acts on affine space, in addition to G, in which case algorithms compute the set of G-translates of I

• whose stabilizer subgroups in T have maximal dimension; or
• that admit a faithful multigrading by \(\mathbb {Z}^r\) of maximal rank r.

Even with no ambient group action given, the final application is an algorithm to

• decide whether a normal projective variety is abstractly toric.

All of these loci in B and subsets of G are constructible.

中文翻译：

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自环境同态后的多项式理想二项式是什么时候？

多项式环\（\ Bbbk [\ mathbf {x}] \）中的理想 I是否可以通过更改坐标而移动到二项式\（\ mathbf {x} ^ \ mathbb生成位置的位置A- \ lambda \ mathbf {x} ^ \ mathbf {b} \）与\（\ lambda \ in \ Bbbk \）或单位二项式（即\（\ lambda = 0 \）或1）？品种可以移到复曲面的位置吗？通过fibering的摹的-translates 我了代数群 摹作用于仿射空间，这些问题都是关于家庭问题的特殊情况 \（\ mathcal {I} \）在任意基地的理想 乙。在此一般设置下的主要结果是找到\（\ mathcal {I} \）的光纤 在B中的点的轨迹的算法 

• 被包含在第二个家庭的纤维 \（\ mathcal {I}'\）超过理想的 乙;
• 至少定义d的各种尺寸 ；
• 由二项式生成；要么
• 由单位二项式生成。

当\（\ mathcal {I} \）的纤维为素数时，还提出了一种更快的遏制算法 。大光纤算法是概率性的，但可能比已知的确定性算法更快。应用包括其中第二组的设定 Ť作用在仿射空间中，除了 G ^，在这种情况下的算法计算所述一组ģ的-translates 我

• T中的稳定剂亚组 具有最大尺寸；要么
• 接受最大等级r的\（\ mathbb {Z} ^ r \）的 忠实多重评分 。

即使没有给出环境小组动作，最终的应用程序还是

• 确定正常的投影变种是否抽象复曲面。

B中的所有这些基因座 和G的子集 都是可构建的。