Multivariate Creative Telescoping

Let us define the algebra

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

where $\boldsymbol{x} = (x_1,\dots,x_n)$.

Let $I$ be a holonomic ideal of $D$ and $p\in D$.

The function MCT returns an operator $L\in \mathbb{Q}(t)\langle \partial_t\rangle$ such that $Lp\in I + \sum_{i=1}^n \partial_i D$. In particular if $I$ is the annihilator of a holonomic function $f$ then $L$ annihilates the integral

\[\int pf \boldsymbol{dx}\]

provided it has natural boundaries.

The signature of the function is the following.

MultivariateCreativeTelescoping.MCT — Function

MCT(spol :: OrePoly, gb :: Vector{OrePoly{T,M}}, A::OreAlg) where {T,M}

source

Assumption:

spol corresponds to the operator $p$
gb is a Gröbner basis of a holonomic subideal of $I$ for the order defined in A
A corresponds to the algebra $D$ and its order eliminates $dt$

It is advised for efficiency to use an order of the form lex $dt$ > grevlex [list of other polynomial variables].

Example

In this example we prove that the integral

\[I(t) = \int_\gamma \frac{x}{x-t}dx\]

where $\gamma$ is a loop satisfies the differential equation $t\partial_t -1$ for any $t$ inside that loop. This reflects the well-known fact $I(t) = 2i\pi t$.

First we load the package.

julia> using MultivariateCreativeTelescoping

We define the algebra.

julia> A = OreAlg(order = "lex dt > grevlex x dx",ratdiffvars=(["t"],["dt"]),poldiffvars=(["x"],["dx"]))
Ore algebra

Then we compute an holonomic ideal included in the annihilator of $1/(x-t)$. For more details see the Weyl closure subsection of the other functionalities section.

julia> ann = [parse_OrePoly("dt*(x-t)",A), parse_OrePoly("dx*(x-t)",A)]
Vector of Ore polynomials

julia> init = weyl_closure_init(A)
WeylClosureInit

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

We can now apply the integration algorithm

julia> LDE = MCT(parse_OrePoly("x",A), gb, A)
Ore polynomial 

julia> prettyprint(LDE,A)
(t)dt + (-1)