## Section: Research Program

### Effective algebra

To develop a computational study of problems coming from control theory, signal processing, and multidisciplinary domains, parts of algebraic theories must be studied within an effective approach: methods and theoretical results must be made algorithmic based on computer algebra techniques appropriated for efficient implementations in computer algebra systems.

#### Polydisc Nullstellensatz & effective version of a theorem of Deligne

The works on stability and stabilization
problems of multidimensional systems, developed in the former ANR
MSDOS (2014–2018),
have shown the importance for developing an effective version of the
module theory over the ring of rational functions without poles in
the closed unit polydisc of ${\u2102}^{n}$ [90], [47].
The stabilizability (resp. the existence of a doubly
coprime factorization) of a multidimensional system is related to a module-theoretical property (projectivity, resp. freeness) that has to be algorithmically verified
prior to compute stabilizing controllers (resp. the standard
Youla-Ku$\stackrel{\u02c7}{\mathrm{c}}$era parametrization of all stabilizing
controllers). Based on the works [89], [90], in
[47],
we have recently proved that the stabilizability
condition is related to the development of an algorithmic proof of
the so-called *Polydisc Nullstellensatz* [49], a natural extension of Hilbert's *Nullstellensatz* for the above-mentioned ring (see e.g. [40]). In addition,
the existence of a doubly coprime factorization is related to a
theorem obtained by (the Fields medalist) Deligne with a
non-constructive proof [90]. This theorem can be seen as an
extension of the famous Quillen-Suslin theorem (Serre's conjecture) [77].
Based on our experience of the first implementation of the
Quillen-Suslin theorem in the computer algebra system (`Maple` )
[62], we aim to develop this effective framework as well as a
dedicated `Maple` package.

#### Effective version of Spencer's theory of formal integrability of PD systems

A differential geometric counterpart of differential algebra and differential elimination theory [101], [76] is the so-called Spencer's theory of formal integrability and involutive PD systems [85], [104]. For linear PD systems, this theory can be seen as an intrinsic approach to Janet or Gröbner bases for noncommutatibe polynomial rings of PD operators. No complete algorithmic study of Spencer's theory has been developed yet. We aim to develop it as well as to implement it. The understanding of the connections between the different differential elimination theories (Janet [102] or Gröbner bases [40], Thomas decomposition [102], differential algebra [101], [76], Spencer's theory [85], exterior differential systems [51], etc.) will also be investigated. On a longer term, applications of Spencer's theory to Lie pseudogroups and their applications in mathematical physics (e.g. variational formulations based on Lie (pseudo)groups) [86] will be investigated and implemented.

#### Rings of integro-differential operators & integro-differential algebra

The main contribution of this axis is the development of effective elimination theories for both linear and nonlinear systems of integro-differential equations.

##### Linear systems of integro-differential equations

The rings of integro-differential operators are more complex than the purely
differential case [96], [97] due to the existence of
zero-divisors or the fact of having a *coherent ring* instead of a
*Noetherian ring* [39]. We want to
develop an algorithmic study of these rings. Following the direction
initiated in [95] for the computation of zero divisors,
we first want to develop algorithms for the computation of
left/right kernels and left/right/generalized inverses of matrices
with entries in such rings, and to use them to develop a module-theoretic approach to linear systems of integro-differential
equations. Following [95],
standard questions addressed within the computer algebra community
such as the computation of
rational/exponential/hyperexponential/etc. solutions will also be
addressed. Moreover, famous Stafford's results [105],
algorithmically studied in [96], [97] for rings of
PD operators, are known to still hold for rings of
integro-differential operators [39]. Their algorithmic
extensions will be investigated and our corresponding implementation will be
extended accordingly. Finally, following [93], [95], an algorithmically study of rings
of integro-differential-delay operators will be further developed
as well as their applications to the equivalence problem of
differential constant/varying/distributed delay systems
(e.g. Artstein's reduction, Fiagbedzi-Pearson's transformation) and
their applications to control theory.

##### Nonlinear systems of integro-differential equations

*Integro-differential algebra* is an extension of Ritt-Kolchin's *differential algebra* [101], [76]
that also includes integral operators. This extension is now attracting more attention in mathematics, computer algebras, and control theory.
This new type of algebras will be algorithmically studied for integro-differential
nonlinear systems. To do that, concepts such as integro-differential
ideals and varieties have to be introduced and studied for developing an integro-differential elimination
theory which extends the current differential elimination theory [45], [46], [72]. A
`Maple` prototype will first be developed and then a `C` library
when experience will be gained.