Clarify when to use ParamPunPam.jl vs Groebner.jl

That's an interesting question !

To quote my scientific advisor, parameters would typically represent some quantities which are assumed to be known. In other words, one would use parametric Groebner bases if one is interested in solving the equations by expressing the variables in terms of parameters.

Formally, the difference between parameters and variables is not large. The choice between ParamPunPam and Groebner will probably depend on the specific task at hand.

To illustrate what I mean, I bring the following example. Say, we want to solve this linear system in $x,y$:

$$ \begin{pmatrix} a & a + b \ a - b & a^2 \end{pmatrix} \times \begin{pmatrix} x\ y \end{pmatrix} \= \begin{pmatrix} b^2\ a^2 \end{pmatrix} $$

We can easily cast this problem into a setting of parametric Groebner bases:

using Nemo, ParamPunPam
R_param, (a, b) = QQ["a", "b"]
R, (x,y) = FractionField(R_param)["x","y"]

paramgb([
    a*x  + (a+b)*y  - b^2,
    (a-b)*x + a^2*y - a^2
])

Running this will give you the solutions in $x$ and $y$, e.g., we get $y = -(-a^3 + a b^2 - b^3)/(a^3 - a^2 + b^2)$.

Now note that the same result can be obtained with Groebner, if we use $\mathbb{Q}[x,y,a,b]$ with $a,b < x,y$, instead of the parametric $\mathbb{Q}(a,b)[x,y]$. Indeed, in this basis

using Nemo, Groebner
R, (x,y,a,b) = QQ["x","y","a","b"]

groebner([
    a*x + (a+b)*y   - b^2,
    (a-b)*x + a^2*y - a^2
])
# returns
3-element Vector{QQMPolyRingElem}:
 y*a^3 - y*a^2 + y*b^2 - a^3 + a*b^2 - b^3
 x*b - y*a^2 + y*a + y*b + a^2 - b^2
 x*a + y*a + y*b - b^2

if we group the terms in the first polynomial, we get $y (a^3 - a^2 + b^2) = -(-a^3 + a b^2 - b^3)$. It is the same result we obtained when using parametric bases, if we divide by $(a^3 - a^2 + b^2)$.

To sum up, parametric Groebner bases and usual Groebner bases are two sides of the same coin; which to use depends on your task and imagination :)

sumiya11 / ParamPunPam.jl

Clarify when to use ParamPunPam.jl vs Groebner.jl #4