Vladimir Gerdt, distinguished scientist in symbolic computation, left us on January 5, 2021. Over 30 years, I met him many times and cooperated with him on various occasions. In honor of his memory, I will recollect some of the wonderful moments he had with us and will brief the history of developments on Thomas decomposition in which he was heavily involved. Vladimir was a strong academic leader with deep thought, broad view, warm heart, and tireless dedication. His research spanned across several disciplines from computer algebra to quantum computation and resulted in over 200 publications. He will forever be remembered for all his contributions to the advance of science and to our community.

Simple decomposition, also called Thomas decomposition, is the process to decompose an algebraic or differential polynomial set into squarefree triangular sets with associated zero and ideal relationships. With his collaborators, Vladimir Gerdt greatly developed the theories, algorithms, implementations, and applications of simple decomposition. In this talk we first review the algorithms for simple decomposition and then report the on-going work of extending the theories and methods of simple decomposition to characteristic decomposition, in which Groebner bases and triangular sets are both integrated.

When Vladimir Gerdt and myself first met in 1997 we discussed the interaction between “theory” and applications in computer algebra, that it could and should be improved, and that we could and should do something about it: Our discussion materialized in the Computer Algebra in Scientific Computing (CASC) conference series, starting out a year later in St. Petersburg. I shall talk a bit about this series which Vladimir chaired and shaped and nourished for about twenty years, and about the (possible) meaning of “theory” in computer algebra as we theorized about it in 1997, and how it developed.

In the computer algebra community, Vladimir Gerdt is best known as a mathematician interested in commutative, differential and difference algebra. But by training, Vladimir was a physicist much interested in constrained mechanical systems and field theories. Such systems automatically appear in many applications - either because a mathematical modelling is easier in redundant coordinates or because of an underlying symmetry. This connection between symmetries and constraints has been fundamental for elementary particle physics for many decades and is intimately connected with Nobel prize laureate Paul Dirac. He developed in the 1940s an approach for the treatment of constrained Hamiltonian systems in the ODE case and applied his methods in an ad hoc manner also to PDEs. The Dirac theory is notoriously subtle and full of pitfalls; the extension to PDEs is far from obvious. Vladimir developed together with collaborators a fully algorithmic version of the Dirac theory for polynomial systems of ODEs and also studied Lagrangian versions of it or the treatment of field theories. The talk will review some of his achievements in this domain.

In this talk, I will give a short history of my research cooperation with Vladimir started in 2011. For this purpose, I give an overview about Gröbner bases and involutive bases and a short introduction about our joint works with him. Then, I will conclude my talk with an important question posed by Vladimir.

A Bar Code is a bidimensional diagram used to encode the properties of (finite) monomial sets. In particular, multiplicative variables and completeness with respect to Janet division can be studied by means of Bar Codes, as well as non-multiplicative powers with respect to Janet-like division. In this talk we will deal with the study of Janet division by means of Bar Codes, and in particular with the study of the following problem: "is there a variable ordering s.t. a given set of terms is complete with respect to Janet division, according to that ordering"? We will give an algorithmic answer to the problem.

When we accepted to give a talk, we were planning to complete our investigation on involutive bases on the Tamari ring $\mathcal{A}:=\mathbf{k}[X,Y]/\mathbb{I}(YX-X^2Y)$ where we have a puzzling example [3, Example 3 d)]: if we consider the ideal $\mathbb{I}(X,Y)\subset{\mathcal A}$ and we assign to $Y$ as multiplicative variables both $X$ and $Y$ we have a trivial division $$\mathbb{I}(X,Y)=\left\{X^aY^b\star Y, (a,b)\in{\mathbb{N}}^2\right\}\sqcup\left\{X^a\star X, a\in{\mathbb{N}}\right\}$$ but if we assign both variables to $X$ we have an infinite decomposition \begin{align*} \mathbb{I}(X,Y)=\left\{X^aY^b\star X, (a,b)\in{\mathbb{N}}^2\right\}\sqcup\left\{Y^b\star Y, b\in{\mathbb{N}}\right\}\\ \bigsqcup \left(\sqcup_{X^{2i+1}Y^j\in W} \left\{Y^b\star X^{2{i+b}+2^b}Y^{b+j}, b\in{\mathbb{N}}\right\}\right) \end{align*} with $W=XY,XY^2,X^3Y^2,XY^3,X^3Y^3,X^5Y^3,X^7Y^3,\ldots$.

Since the example seems to be a version of classical Pommaret division, we were hoping to adapt the related classical results to this example, but it was not yet clear to us how to adapt the classical completion procedure.

In the meantime, Wolfgang Rump, in connection with [8], posed us the following question:

Consider the ring with generators $p$ and $q,q'$ with relations $qq'=q'q=1$ and $pq-qp=p^2$. Thus $q'=q^{-1}$ [$\cdots$]. One easily shows that $(q+p)(q^{-1}-pq^{-2})=1$. My question: Is $q+p$ invertible? Does $px=0$ imply that $x=0$?

Since the classical techniques of Zacharias' canonical representation [9] seemed to be able to give a fast and satisfactory answer to these questions, we dealt with this problem before coming back to our investigation. Thus we made the unjustifiable ridiculous mistake of assuming that the given basis was Gröbner, while as Rump remarked "note that $pq^2-q^2p=2pqp$ reduces $p^3=(pp)p=p(pp)$ in two ways".

This gives a first intriguing Ufnarovski-like sequence [4, 5] with coefficients in $\mathbb{Z}$: $G:=\left\{f_i : i\in{\mathbb N}\setminus\{0\}\right\}$ with \begin{align*} & f_1=p^2-pq+qp, f_2=2pqp-pq^2+q^2p, \\ & f_3 =3pq^2p-pq^3+q^3p, \ldots,f_n =npq^{n-1}p-pq^n+q^np,\ldots; \end{align*} it would be just sufficient to consider this sequence under a suitable term-ordering $<$ on $\langle p,q\rangle$, satisfying $$\deg_p(\tau_1) < \deg_p(\tau_2) \; \Longrightarrow \; \tau_1<\tau_2$$ and discuss the posed questions in this Ufnarovski-like setting, as we will do, to show the power of Zacharias' results for a Buchberger Theory (and practice) of effective associative rings [1, 2, 6, 7].

However, the principal ideal ${\mathbb I}(p^2-pq+qp)\subset {\mathbb Z}\langle p,q\rangle$ introduced by Rump [8] has a more complex Gröbner basis as can be shown by the first most elementary S-polynomials: $$\begin{array}{rcl} f_2qp-qpf_2&\rightarrow& {\bf -2pq^3p}+pqpq^2+q^2pqp=:g\cr f_4+2g&\rightarrow& 0\cr pg+2f_1q^{3}p&\rightarrow&pq^2pq^2-qpqpq^2+{\bf pq^2pqp}\cr gp+2pq^{3}f_1&\rightarrow&pqpq^2p+q^2pqpq-{\bf q^2pq^2p} \cr \end{array}$$ References
[1] Ceria, M., Mora, T.(2017). Buchberger–Zacharias theory of multivariate Ore extensions. Journal of Pure and Applied Algebra 221(12), 2974–3026.
[2] Ceria, M., Mora, T. (2017). Buchberger–Weispfenning theory for effective associative rings. Journal of Symbolic Computation, 83, 112-146.
[3] Ceria, M., Mora, T. (2020). Toward involutive bases over effective rings. Applicable Algebra in Engineering, Communication and Computing, 31(5), 359-387.
[4] Cojocaru S., Ufnarovski V. (1995). Noncommuatative Gröbner basis, Hilbert series, Anick's resolution and BERGMAN under MS-DOS, Computer Science Journal of Moldova 3, 24–39.
[5] Green, E., Mora, T., Ufnarovski, V. (1998). The non-commutative Gröbner freaks. In: Symbolic rewriting techniques. Birkhäuser, Basel, pp. 93-104.
[6] Mora, T. (2020). Zacharias representation of effective associative rings. Journal of Symbolic Computation, 99, 147-188.
[7] Nguefack, B., Pola, E. (2019). Effective Buchberger-Zacharias-Weispfenning theory of skew polynomial extensions of restricted bilateral coherent rings. J. Symb. Comp.
[8] Rump W. (2021) Degenerate involutive set-theoretic solutions to the Yang-Baxter equation, preprint.
[9] Zacharias, G. (1978). Generalized Gröbner bases in commutative polynomial rings. Bachelor Thesis, MIT Dept. of Comp. Sci, 147-160.