@phdthesis{Walsh2024, author = {Walsh, Florian}, title = {Computing the Binomial Part of Polynomial Ideals}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus4-15096}, school = {Universit{\"a}t Passau}, pages = {vi, 131 Seiten}, year = {2024}, abstract = {Given an ideal in a polynomial ring over a field, we present a complete algorithm to compute its binomial part.}, language = {en} } @article{KreuzerWalsh2024, author = {Kreuzer, Martin and Walsh, Florian}, title = {Computing the binomial part of a polynomial ideal}, series = {Journal of Symbolic Computation (Online ISSN: 1095-855X)}, volume = {2024}, journal = {Journal of Symbolic Computation (Online ISSN: 1095-855X)}, number = {124}, publisher = {Elsevier}, address = {Amsterdam}, doi = {10.1016/j.jsc.2024.102298}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus4-18708}, pages = {30 Seiten}, year = {2024}, abstract = {Given an ideal I in a polynomial ring K[x1,...,xn] over a field K, we present a complete algorithm to compute the binomial part of I, i.e., the subideal Bin(I) of I generated by all monomials and binomials in I. This is achieved step-by-step. First we collect and extend several algorithms for computing exponent lattices in different kinds of fields. Then we generalize them to compute exponent lattices of units in 0-dimensional K-algebras, where we have to generalize the computation of the separable part of an algebra to non-perfect fields in characteristic p. Next we examine the computation of unit lattices in finitely generated K-algebras, as well as their associated characters and lattice ideals. This allows us to calculate Bin(I) when I is saturated with respect to the indeterminates by reducing the task to the 0-dimensional case. Finally, we treat the computation of Bin(I) for general ideals by computing their cellular decomposition and dealing with finitely many special ideals called (s,t)-binomial parts. All algorithms have been implemented in SageMath.}, language = {en} } @article{KreuzerMiasnikovWalsh2024, author = {Kreuzer, Martin and Miasnikov, Alexei and Walsh, Florian}, title = {Decomposing finite Z-algebras}, series = {Journal of Algebra}, volume = {2025}, journal = {Journal of Algebra}, number = {664 B}, publisher = {Elsevier}, address = {Amsterdam}, issn = {1090-266X}, doi = {10.1016/j.jalgebra.2024.10.027}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus4-19184}, pages = {206 -- 246}, year = {2024}, abstract = {For a finite Z-algebra R, i.e., for a ring which is not necessarily associative or unitary, but whose additive group is finitely generated, we construct a decomposition of R/Ann(R) into directly indecomposable factors under weak hypotheses. The method is based on constructing and decomposing a ring of scalarsS, and then lifting the decomposition ofSto the bilinear map given by the multiplication of R, and finally to R/Ann(R). All steps of the construction are given as explicit algorithms and it is shown that the entire procedure has a probabilistic polynomial time complexity in the bit size of the input, except for the possible need to calculate the prime factorization of an integer. In particular, in the case when Ann(R)=0, these algorithms compute direct decompositions of R into directly indecomposable factors.}, language = {en} }