Comptes Rendus
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Combinatorics
Finiteness of rank for Grassmann convexity
Nicolau Saldanha1 ; Boris Shapiro2; Michael Shapiro3, 4
1 Departamento de Matemática, PUC-Rio, R. Mq. de S. Vicente 225, Rio de Janeiro, RJ 22451-900, Brazil
2 Department of Mathematics, Stockholm University, SE-106 91, Stockholm, Sweden
3 Department of Mathematics, Michigan State University, East Lansing, MI 48824-1027, USA
4 National Research University Higher School of Economics, Moscow, Russia
Comptes Rendus. Mathématique, Volume 361 (2023), pp. 445-451.

The Grassmann convexity conjecture, formulated in [8], gives a conjectural formula for the maximal total number of real zeroes of the consecutive Wronskians of an arbitrary fundamental solution to a disconjugate linear ordinary differential equation with real time. The conjecture can be reformulated in terms of convex curves in the nilpotent lower triangular group. The formula has already been shown to be a correct lower bound and to give a correct upper bound in several small dimensional cases. In this paper we obtain a general explicit upper bound.

La conjecture sur la convexité du Grassmannien formulée dans [8] suggère une formule pour le nombre total maximal de zéros réels des Wronskiens consécutifs d’une solution fondamentale arbitraire d’un système disconjugué d’équations différentielles ordinaires linéaires à temps réel. La conjecture peut être formulée en termes de courbes convexes dans le groupe nilpotent triangulaire inférieur. Il a déjà été prouvé que la formule donne une borne inférieure correcte et que dans plusieurs cas de basse dimension, elle donne la borne supérieure correcte. Dans cet article nous obtenons une borne supérieure explicite générale.

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DOI: 10.5802/crmath.343
Classification: 34C10, 05B30, 57N80

Nicolau Saldanha 1; Boris Shapiro 2; Michael Shapiro 3, 4

1 Departamento de Matemática, PUC-Rio, R. Mq. de S. Vicente 225, Rio de Janeiro, RJ 22451-900, Brazil
2 Department of Mathematics, Stockholm University, SE-106 91, Stockholm, Sweden
3 Department of Mathematics, Michigan State University, East Lansing, MI 48824-1027, USA
4 National Research University Higher School of Economics, Moscow, Russia
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Nicolau Saldanha; Boris Shapiro; Michael Shapiro. Finiteness of rank for Grassmann convexity. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 445-451. doi : 10.5802/crmath.343. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.343/

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[7] Nicolau Saldanha; Boris Shapiro; Michael Shapiro Grassmann convexity and multiplicative Sturm theory, revisited, Mosc. Math. J., Volume 21 (2021) no. 3, pp. 613-637 | DOI | Zbl

[8] Boris Shapiro; Michael Shapiro Projective convexity in P 3 implies Grassmann convexity, Int. J. Math., Volume 11 (2000) no. 4, pp. 579-588 | DOI | Zbl

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