Constructing Gröbner bases for Noetherian rings

Abstract

We prove constructively that every finitely generated polynomial ideal has a Gröbner basis provided that the ring of coefficients is Noetherian in the sense of Richman and Seidenberg. That is, we give a constructive termination proof for a variant of the otherwise well-known algorithm to compute the Gröbner basis. In combination with a purely order-theoretic result we have proved in a separate paper, this yields a unified constructive proof of the Hilbert basis theorem for all Noether classes: if a ring belongs to a Noether class, then so does the polynomial ring. Our proof can be seen as a constructive rereading of one of the classical proofs, in the spirit of the partial realisation of Hilbert's programme in algebra put forward by Coquand and Lombardi. The rings under consideration need not be commutative, but are understood to be coherent, and strongly discrete: that is, they admit a membership test for every finitely generated ideal. As a complement we provide a prime decomposition for commutative rings possessing the finite-depth property.

Syntax for Semantics: Krull’s Maximal Ideal Theorem

“…Several constructive notions of Noetherianity have been proposed in the literature (Richman 1974;Seidenberg 1974;Mines, Richman, and Ruitenburg 1988;Jacobsson and Löfwall 1991;Coquand and Persson 1998a;Perdry 2004;Perdry and Schuster 2011;Blechschmidt 2017), the litmus test usually being Hilbert's Basis Theorem. Martin-Löf has proposed a definition replacing the Ascending Chain Condition by an induction principle (Jacobsson and Löfwall 1991), to be recalled in the following.…”

Finite Methods in Mathematical Practice

“…. , T n ] over a field; see also [100,102,120,132]. Also, to prove with constructive means the termination of Buchberger's algorithm for the computation of Gröbner bases, which is one of the cornerstones of computer algebra [1], it suffices that any K[T 1 , .…”

“…Understand "R is Noetherian" as the classically equivalent finitedepth property [101,102]: i.e., every tree whose nodes are labelled by finitely generated ideals of R has finite depth provided that along every branch of the tree the ideals labelling the nodes form an ascending sequence. 51 51 Alternatively, one can strengthen "R is Noetherian" to "R is strongly Noetherian" [99].…”

“…Under these additional hypotheses one can indeed [99,102] grow a binary tree with root labelled by √ 0 which is labelled strictly increasingly by finitely generated ideals -and which therefore is finite -and whose leaves are labelled by finitely generated prime ideals p 1 , . .…”