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This paper is concerned with linear algebra based methods for solving exactlypolynomial systems through so-called Gr\"obner bases, which allow one tocompute modulo the polynomial ideal generated by the input equations. This is atopical issue in non-linear algebra and more broadly in computationalmathematics because of its numerous applications in engineering and computingsciences. Such applications often require geometric computing features such asrepresenting the closure of the set difference of two solution sets to givenpolynomial systems. Algebraically, this boils down to computing Gr\"obner basesof colon and/or saturation polynomial ideals. In this paper, we describe andanalyze new Gr\"obner bases algorithms for this task and presentimplementations which are more efficient by several orders of magnitude thanthe state-of-the-art software.

New efficient algorithms for computing Gr\"obner bases of saturation ideals (F4SAT) and colon ideals (Sparse-FGLM-colon)