I know of the famous Buchberger algorithm to find Gröbner bases. However I am curious if it is not somehow possible to compute them using only linear algebra.
Own Work: The little I know so far is that
- It is possible to represent multiplication of polynomials as convolutions of their coefficients. 2D convolution if 2-variable polynomial, 3D convolution if 3-variable et.c.
- Then I know we can represent convolution of 1D vectors by multiplication by Toeplitz matrices, and my suspicions are that we should be able to do the same for higher number of dimensions..
Should not this provide at least a tool to set up equation systems governing products of multivariate polynomials?