2024/03/11 by Martin Kreuzer, Lorenzo Robbiano, Kreuzer, Martin +1
Computer Science · Mathematics · #Polynomial and algebraic computation #Commutative Algebra and Its Applications #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2403.06415
Let K be a field and P=K[x1,…,xn]. The technique of elimination by substitution is based on discovering a coherently Z=(z1,…,zs)-separating tuple of polynomials (f1,…,fs) in an ideal I, i.e., on finding polynomials such that fi = zi - hi with hi ∈ K[X ∖ Z]. Here we elaborate on this technique in the case when P is non-negatively graded. The existence of a coherently Z-separating tuple is reduced to solving several P0-module membership problems. Best separable re-embeddings, i.e., isomorphisms P/I \longrightarrow K[X ∖ Z] / (I ∩ K[X ∖ Z]) with maximal #Z, are found degree-by-degree. They turn out to yield optimal re-embeddings in the positively graded case. Viewing P0 \longrightarrow P/I as a fibration over an affine space, we show that its fibers allow optimal Z-separating re-embeddings, and we provide a criterion for a fiber to be isomorphic to an affine space. In the last section we introduce a new technique based on the solution of a unimodular matrix problem which enables us to construct automorphisms of P such that additional Z-separating re-embeddings are possible. One of the main outcomes is an algorithm which allows us to explicitly compute a homogeneous isomorphism between P/I and a non-negatively graded polynomial ring if P/I is regular.