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

自从Buchberger[1]引入Gröbner基以来，Gröbner基在算法代数中起着至关重要的作用。然而，即使算法和计算机有所改进，他们的计算通常对于大型例子来说也是不可行的。因此，重要的是确定可以有效地进行Gröbner基计算的理想类，并进一步发展已知的算法。Gröbner基的关键参数是其多项式的最大次数，这可以从所有已知的复杂性上界的证明中看出。因此，这个参数将针对实践中经常出现的不同类别的理想(例如，激进理想、质数理想和环面理想)来处理。由此得到的见解应应用于增强计算Gröbner基的算法或类似的算法。

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.