# Workshop in Honor of Vladimir Gerdt

## Sunday, 18 July 2021

With great sadness have we learned that our dear colleague and friend Vladimir Gerdt passed away early January, 2021.

We would like to commemorate Vladimir's work and enthusiasm in a workshop preceding the ISSAC tutorials.

The workshop will take place in the afternoon of Sunday 18 July 2021 (Saint Petersburg time, UTC +3) on Zoom.

Preliminary Schedule:

Time Speaker Title / Abstract
13:45 Welcome
13:50 Dongming Wang [+] Thomas Decomposition: A Brief History. In memory of Vladimir Gerdt
Abstract:

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.

14:00 Chenqi Mou TBA
14:30 Ernst W. Mayr [+] Theory in CA$\subseteq$SC?
Abstract:

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.

15:00 break
15:15 Dmitry Lyakhov On the Algorithmic Linearizability of Nonlinear Ordinary Differential Equations
15:45 Werner M. Seiler [+] Vladimir Gerdt's Work on Constrained Dynamics
Abstract:

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.

16:15 Amir Hashemi [+] The history of my research cooperation with Vladimir
Abstract:

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.

16:45 break
17:00 Michela Ceria [+] Applications of Bar Code to involutive divisions and a greedy algorithm for complete sets
Abstract:

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.

17:30 Teo Mora [+] De Nugis Groebnrialium 6: Rump, Ufnarovski, Zacharias
Abstract:

Recently, Wolfgang Rump, in connection with [10], 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 $x=0$?

We easily gave an answer to all questions using the classical tecniques of Zacharias' canonical representation [11]. Unfortnately we made the unjustifiable ridiculous mistake of assuming that the given basis were Gröbner, while as Rump remarked "note that $pq^2-q^2p=2pqp$ reducing $p^3=(pp)p=p(pp)$ in two ways". This gives a first intriguing Ufnarovski-like sequence [3, 4, 5, 6, 7] 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 any term-ording on $\langle p,q,q^{-1}\rangle$ for which \begin{align}%\label{Eq1} \deg_p(\tau_1) < \deg_p(\tau_2) \; \Longrightarrow \; \tau_1 < \tau_2, \mbox{ for each } \tau_1, \tau_2\in\langle p,q,q^{-1}\rangle \end{align} 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, 8, 9].

However, the principal ideal ${\mathbb I}(p^2-pq+qp)\subset {\mathbb Z}\langle p,q\rangle$ introduced by Rump [10] has a more complex Gröbner basis under the term-ordering on $\langle p,q\rangle$, where $\prec$ denotes the lexicographial order with $p\prec q$, \begin{align}%\label{Eq1} &\tau_1<\tau_2 \iff \deg_p(\tau_1) < \deg_p(\tau_2) \mbox{ or } \deg_p(\tau_1) = \deg_p(\tau_2)\\& \nonumber \mbox{ and } \tau_1 \prec \tau_2, \mbox{ for each } \tau_1, \tau_2\in\langle p,q\rangle \end{align} as can be shown by the first most elementary S-polynomials: $$\begin{array}{rcl} f_2qp-qpf_2&\rightarrow& {\bf -2pq^3p}+pqpq^2+q^2pqp=:g\cr f_4+2g&\rightarrow& 0\cr pg+2f_1q^{3}p&\rightarrow&pq^2pq^2-qpqpq^2+{\bf pq^2pqp}\cr gp+2pq^{3}f_1&\rightarrow&{\bf pqpq^2p}+q^2pqpq-q^2pq^2p \cr \end{array}$$ References
[1] Ceria, M., Mora, T.(2017). Buchberger–Zacharias theory of multivariate Ore extensions. J. Pure Appl. 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] 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.
[4] Green, E., Mora, T., Ufnarovski, V. (1998). The non-commutative Gr\"obner freaks. In Symbolic rewriting techniques (pp. 93-104). Birkhäuser, Basel.
[5] Månsson, J. (2001). A Prediction Algorithm for Rational Language. Licentiate Thesis, Lund University.
[6] Månsson, J., Nordbeck, P. (2005). A generalized Ufnarovski graph. Applicable Algebra in Engineering, Communication and Computing, 16(5), 293-306.
[7] Mårtensson, K. (2006). An algorithm to detect regular behaviour of binomial Gröbner Basis rational language. Master's Thesis, Lund University.
[8] Mora, T. (2020). Zacharias representation of effective associative rings. Journal of Symbolic Computation, 99, 147-188.
[9] Nguefack, B., Pola, E. (2019). Effective Buchberger-Zacharias-Weispfenning theory of skew polynomial extensions of restricted bilateral coherent rings. J. Symb. Comp.
[10] Rump W. (2021) Degenerate involutive set-theoretic solutions to the Yang-Baxter equation, preprint.
[11] Zacharias, G. (1978). Generalized Gröbner bases in commutative polynomial rings. Bachelor Thesis, MIT Dept. of Comp. Sci, 147-160.

18:00 Daniel Robertz Remembering collaborations with Vladimir Gerdt