Polish-Italian mini-workshop

Apolarity in Varsavia (AIVAR)

Mon 20 - Thu 23 January 2025

IMPAN, Warsaw, Poland

Overview

The collaboration between Polish and Italian algebraic geometers has a long standing tradition. Within this miniworkshop we focus on apolarity methods in geometry. We will not have any formal talks, instead we will focus on semi-organised discussions and loose flow of thoughts between the participants. This formula will promote research advancement and hopefully result in a scientific discovery.

Previous events in Warsaw in a similar spirit include Tensor days (2021), discussion sessions during the AGATES semester (2022) and its workshops, and secant varieties working groups 2018-2024.

Participants:

• Weronika Buczyńska (Math @ University of Warsaw)
• Alessandra Bernardi (University of Trento)
• Jarosław Buczyński (IMPAN, Warsaw)
• Stefano Canino (Math @ University of Warsaw)
• Cosimo Flavi (Math @ University of Warsaw)
• Francesco Galuppi (Math @ University of Warsaw)
• Maciej Gałązka (University of Trento)
• Vincenzo Isoldi (University of Trento)
• Jakub Jagiełła (Math @ University of Warsaw)
• Joachim Jelisiejew (Math @ University of Warsaw)
• Yuze Luan (IMPAN, Warsaw)
• Giovanni Moreno (Physics @ University of Warsaw)
• Weronika Obcowska (Math @ University of Warsaw)
• Piotr Oszer (Math @ University of Warsaw)
• Paweł Pielasa (Math @ University of Warsaw)
• Oriol Reig (University of Trento)
• Prajwal Samal (IMPAN, Kraków)

Organisers:

• Jarosław Buczyński, Institute of Mathematics of Polish Academy of Sciences (IMPAN, Warsaw);
• Maciej Gałązka (University of Trento)
• Francesco Galuppi (University of Warsaw)

Institutional organiser:

• Banach Center, Warsaw

Location:

Daily schedule

Main topics:

1. Geometric decomposition algorithm
2. Schur apolarity revisited

The first topic builds on the decomposition algorithm originating from the works Brachat, Comon, Mourrain, and Tsigaridas. Any homogeneous polynomial of degree d can be written as a sum of d-th powers of linear forms. It is an important problem to find a decomposition into such sum with a minimal number of summands. The algorithm of Brachat, Comon, Mourrain, and Tsigaridas finds all such decompositions. In practice, the algorithm is efficient for nice polynomials, but formally, in general, the complexity is rather depressing. Moreover, the algorithm is generalised to other objects including decompositions of tensors into sums of simple tensors.
We reinterpreted the algorithm in geometric terms using apolarity techniques. Moreover, investigated if the algorithm can be generalised to search for border decompositions (or at least weak border decompositions) by merging it with border apolarity technique.

Within the the second topic we revisited the so called "Schur apolarity" introduced by Staffolani and searched for loose threads to pull.

Notes:

Funding

• grant from NCN,
• "Tensor rank and its applications to signature tensors of paths", project 2023/51/D/ST1/02363, awarded by the National Science Center, Poland.

Ajwar picture from wikipedia, other pictures by Buczyńscy.