Degree bounds and synchronization in Gröbner basis computations for affine semi-regular systems

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Determining the complexity of computing Gröbner bases is an important problem in both theory and practice, and solving degrees provide a central measure of this complexity. We study solving degrees and Gröbner basis computations for affine polynomial systems, with particular emphasis on semi-regular sequences. We first derive two upper bounds for the maximum Gröbner basis degree of the homogenized system. One is based on a regular initial subsequence of the highest-degree homogeneous parts.