Gröbner bases

This package implements two algorithms to compute Gröbner bases, namely F4 and F5. The F5 implementation is based on Lairez2024. It is recommended to use the F4 implementation as it is faster.

f4(gens :: Vector{OrePoly{T,M}}, A :: Alg)

f5(gens :: Vector{OrePoly{T,M}}, A :: Alg)

Example

julia> using MultivariateCreativeTelescoping

julia>  A = OreAlg(order="grevlex x y > grevlex dx dy",poldiffvars=(["x","y"],["dx","dy"]))
Ore algebra

julia> g = parse_vector_OrePoly("[dx - 2*x, dy + 3*y^2]",A)
Vector of Ore polynomials
    
julia> gb1 = f4(g,A)
Vector of Ore polynomials

julia> gb2 = f5(g,A)
Vector of Ore polynomials

julia> gb1 == gb2
true 

julia> prettyprint(gb1,A)
vector of 2 OrePoly
(1)y^2 + (1//3)dy
(1)x + (-1//2)dx

Weyl closure

Let $D$ be the algebra

\[ D = \mathbb{Q}(\boldsymbol{t})[\boldsymbol{x}]\langle \partial_{\boldsymbol{t}}, \partial_{\boldsymbol{x}}\rangle\]

where $\boldsymbol{t} =(t_1,\dots,t_m)$ and $\boldsymbol{x} = (x_1,\dots,x_n)$.

Let $R$ be the same algebra as with rational coefficients

\[R = \mathbb{Q}(\boldsymbol{t},\boldsymbol{x})\langle \partial_{\boldsymbol{t}}, \partial_{\boldsymbol{x}}\rangle.\]

The Weyl closure of a submodule $S$ of $D^r$ for $r\in\mathbb{N}$ is defined to be the $D$-module $R\cdot S\cap D^r$. This definition generalizes the definition of Tsai given in his PhD thesis which corresponds to the case $m=0$.

Assuming that $D^r/S$ has finite holonomic rank and that generators gens of $S$ are given as input of the function weyl_closure, it returns a Gröbner basis of a $D$-module $S'$ such that $D^r/S'$ is holonomic and $S' \subset R\cdot S\cap D^r$. It is not guaranteed that the algorithm returns the full weyl closure.

weyl_closure_init(A :: OreAlg)

weyl_closure(gens::Vector{OrePoly{T,M}}, A :: OreAlg, init :: WeylClosureInit)

Example

Let $W_2 = \mathbb{Q}[x,y]\langle \partial_t, \partial_x\rangle$ be the second Weyl algebra and let $f = \frac{1}{x^2 -y^3}$. The function $f$ is annihilated by the operators $\partial_x (x^2-y^3)$ and $\partial_y (x^2-y^3)$. But these two equations do not generate the full annihilator of $f$ in $W_2$. A third equation can be found using the Weyl closure.

First we load the package, define the algebra and the two obvious equations annihilating $f$.

julia> using MultivariateCreativeTelescoping

julia> A = OreAlg(order = "grevlex x y > grevlex dx dy",poldiffvars=(["x","y"],["dx","dy"]))
Ore algebra

julia> gens = [parse_OrePoly("dx*(x^2-y^3)",A),parse_OrePoly("dy*(x^2-y^3)",A)]
Vector of Ore polynomials 

Then we can compute its weyl closure

julia> init = weyl_closure_init(A)
WeylClosureInit

julia> gb = weyl_closure(gens,A,init)
Vector of Ore polynomials 

And indeed we found a third equation.

julia> prettyprint(f5(gens,A),A)
vector of 2 OrePoly
(1)dxy^3 + (-1)dxx^2
(1)dyy^3 + (-1)dyx^2

julia> prettyprint(gb,A)
vector of 3 OrePoly
(1)dyy^3 + (-1)dyx^2
(3)dxy^2 + (2)dyx
(3)dxx + (2)dyy + (1)