Comptes Rendus
Combinatoire
Finiteness of rank for Grassmann convexity
[Finitude de rang pour la connexité du Grassmannien]
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.

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.

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.

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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
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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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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[3] Victor Goulart; Nicolau Saldanha Combinatorialization of spaces of nondegenerate spherical curves (2018) (https://arxiv.org/abs/1810.08632)

[4] Victor Goulart; Nicolau Saldanha Stratification by itineraries of spaces of locally convex curves (2019) (https://arxiv.org/abs/1907.01659v1)

[5] Victor Goulart; Nicolau Saldanha Locally convex curves and the Bruhat stratification of the spin group, Isr. J. Math., Volume 242 (2021) no. 2, pp. 565-604 | DOI | Zbl

[6] Philip Hartmann On disconjugate differential equations, Trans. Am. Math. Soc., Volume 134 (1968) no. 1, pp. 53-70 | DOI | Zbl

[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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