Degree Bounds for Gröbner Bases of Important Classes of Polynomial Ideals and Efficient Algorithms (GBiC PolyA)

自Buchberger [1]引入以来，Gröbner基在算法代数中起着至关重要的作用。然而，即使算法和计算机得到改进，它们的计算对于大的例子也是不可行的。因此，重要的是要确定类的理想的计算Gröbner基地可以有效地完成，并进一步发展已知的算法。Gröbner基的关键参数是它们的多项式的最大次数，这可以从所有已知的复杂性界限的证明中看出。因此，这个参数将被视为不同类的理想经常出现在实践中（例如，激进的理想，素理想，环面理想）。将应用由此产生的见解，以增强计算Gröbner基等的算法。

Since their introduction by Buchberger [1], Gröbner bases play a vital role in algorithmic algebra. However, often their calculation is infeasible for large examples, even though algorithms and computers improved. Thus, it is important to identify classes of ideals for which the calculation of Gröbner bases can be done efficiently, and to further develop known algorithms. The crucial parameter of Gröbner bases is the maximal degree of their polynomials as can be seen from all known proofs of complexity bounds. Hence, this parameter will be treated for different classes of ideal often occuring in practice (e.g. radical ideals, prime ideals, and toric ideals). The resulting insights shall be applied in order to enhance algorithms computing Gröbner bases and alike.