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Semigroup ideals and linear Diophantine equations. (English) Zbl 0939.20061
Let $S$ be a finitely generated commutative cancellative monoid, and let $\{n_1,\dots,n_r\}\subset S$ be a set of generators for $S$. Let $k$ be a field, $R=k[S]$ the associated semigroup $k$-algebra, $R=k[X_1,\dots,X_r]$ the polynomial ring, and $\varphi\colon R\to k[S]$ the $k$-algebra homomorphism given by $\varphi(X_i)=n_i$. The author gives a purely algebraic algorithm to calculate a finite set of generators for $\ker\varphi$. As an application, using Gröbner bases, an algorithm is given to determine whether a linear system of equations with integer coefficients having some of the equations in congruences admits non-negative integer solutions.

20M14Commutative semigroups
20M05Free semigroups, generators and relations, word problems
13P10Gröbner bases; other bases for ideals and modules
11D04Linear diophantine equations
68W30Symbolic computation and algebraic computation
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