How beneficial are Gröbner bases for solving systems of equations

Question

So I have understood that Gröbner bases are a pretty big deal and have variety of applications. Namely, if you can represent something with a system of polynomial equations you have the Gröbner bases and anything related to them at your disposal. However, what is the relation of Gröbner basis to other available methods? Is it more like so that any system of polynomial equations can always be solved with a Gröbner bases, but computationally might not be the best possible choice; however alternatives to the G-bases might not provide solution at all?
I am trying to understand how Gröbner bases fit into the bigger picture of optimization and solving problems efficiently.

Michael Morrow · Accepted Answer

There are several improvements you can make on Buchberger's Algorithm for computing Groebner bases, to the point where using other methods usually doesn't make sense. Most computer algebra systems use Groebner bases for their computations since they typically outperform any other method. I think one main reason Groebner bases are so revered is that they provide constructive solutions for many important problems. Take Hilbert's Syzygy Theorem for example. You can prove it using abstract nonsense homological algebra stuff, or you can prove it using Groebner bases. The Groebner basis approach actually gives a recipe for an algorithm to compute finite free resolutions, which is vastly more desirable.

Correct answer by Michael Morrow on December 13, 2020

How beneficial are Gröbner bases for solving systems of equations

One Answer

Add your own answers!

Related Questions

Is cardinality of sigma-algebra for two independent random variables is the sum of cardinalities of two sigma-algebras?

expectation of random probability measure

How many solutions are there to $x_1 + x_2 + x_3 + x_4 = 30$ s.t. $x_1 + x_2 le 20$ and $x_3 ge 7$?

How to prove that this series of functions is uniformly convergent?

Proving that holomorphic map $f : mathbb{C}^n to mathbb{C}^m$ maps holomorphic tangent space at point $p$ to holomorphic tangent space at $f(p)$

Let $X$ be a metric space without isolated points. Then the closure of a discrete set in $X$ is nowhere dense in $X$.

Fundamental theorem of Riemannian Geometry question

If given two groups $G_1,G_2$ of order $m$ and $n$ respectively then the direct product $G_{1}times G_{2}$ has a subgroup of order $m$.

Show that if ${s_{n_k}}_k$ converges to $L$, then ${s_n}_n$ converges to $L$.

Understanding roots of complicated equations

How many solutions does the equation have? (Inclusion-exclusion principle)

Find the eigenspaces corresponding respectively to each eigenvector

Order of a Module

Intuition behind the extected value formula, for continuous cases

Proof that the set $mathbb{Q}left[sqrt2right]$ is an $mathbb{Q}$-vector space

How can we find A using finite difference method?

Probability question about picking $2$ types of balls out of $3$

Let $def#1{mathbf{#1}}{u},{v},{w}inmathbb{R}^2$ be noncollinear. Show that ${x}= r{u}+s{v}+t{w}$ where $r+s+t=1$.

Retract of $F_2$ onto $[F_2,F_2]$

Terence Tao Analysis I on defining subtraction operation problem

Ask a Question