The F4 algorithm re-imagines Buchberger’s algorithm as the row reduction of a Macaulay matrix: each row corresponds to a polynomial; each reduction of one row by another corresponds to one step of reducing an S-polynomial; and any row that completes reduction with a new pivot position corresponds to a new element of the basis. On the other hand, each column corresponds to a term, so that while it is common in linear algebra to exchange a matrix’s columns during row reduction, this has not been done in F4-style algorithms, as it runs the risk of producing an incorrect result. We show that it is possible to adapt an analogous, “dynamic” technique for Buchberger-style algorithms to F4-style algorithms, and we examine its behavior on some commonly referenced benchmark ideals.

中文翻译：

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计算 Gröbner 基的动态 F4 算法

F4 算法将 Buchberger 算法重新想象为麦考莱矩阵的行约简：每一行对应一个多项式；每减少一行对应于减少 S 多项式的一步；并且任何以新的枢轴位置完成归约的行都对应于基础的一个新元素。另一方面，每一列对应一个项，因此虽然在线性代数中在行缩减期间交换矩阵的列是很常见的，但在 F4 风格的算法中并没有这样做，因为它冒着产生不正确的风险结果。我们表明可以将 Buchberger 风格算法的类似“动态”技术应用于 F4 风格算法，并且我们在一些常用参考基准理想上检查其行为。