A zero-dimensional valuation ring is 1-Gröbner

Commutative von Neumann regular rings are 1-Gröbner

“…According to [7], a ring R is 1-Gröbner if for an ideal I ◁ R[X] that is finitely generated, there is a Gröbner basis for I. In [6], the authors prove that a valuation ring of dimension zero is 1-Gröbner. A von Neumann regular ring is another kind of zero-dimensional ring and here we prove that it is also 1-Gröbner, if it is commutative.…”

Rationalizing Denominators Using Gröbner Bases

“…And it has been implemented in many computational softwares including Singular, Maple, CoCoA, Mathematica, Macaulay 2, etc. Many fundamental problems in commutative algebra, computational algebraic number theory, algebraic geometry, graph theory, image processing, cryptography and encoding, and science and engineering can be solved by it algorithmically [13][14][15][16][17][18][19][20][21][22][23]. And the minimal polynomial can be obtained by the reduced Gröbner basis algorithm easily.…”

Global Existence and Extinction Singularity for a Fast Diffusive Polytropic Filtration Equation with Variable Coefficient

“…Problem (1) can be used to describe the compressible fluid flows in a homogeneous isotropic rigid porous medium with u(x, t) being the density of the fluid and α(x) � |x| − s acting as the volumetric moisture content. On the other hand parabolic models like (1), together with differential equation models, stochastic differential equations, and linear systems, are regarded as the powerful tools to solve lots of problems from control engineering, image processing, and other areas (see [4][5][6][7][8]). Because of the degeneracy and the singularity, problem (1) might not have classical solution in general, and hence, we introduce definition of the weak solution as follows.…”