[Finitude de rang pour la connexité du Grassmannien]
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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Saldanha, Nicolau 1 ; Shapiro, Boris 2 ; Shapiro, Michael 3 , 4
CC-BY 4.0
@article{CRMATH_2023__361_G2_445_0,
author = {Saldanha, Nicolau and Shapiro, Boris and Shapiro, Michael},
title = {Finiteness of rank for {Grassmann} convexity},
journal = {Comptes Rendus. Math\'ematique},
pages = {445--451},
year = {2023},
publisher = {Acad\'emie des sciences, Paris},
volume = {361},
number = {G2},
doi = {10.5802/crmath.343},
language = {en},
url = {https://numdam.org/articles/10.5802/crmath.343/}
}
TY - JOUR AU - Saldanha, Nicolau AU - Shapiro, Boris AU - Shapiro, Michael TI - Finiteness of rank for Grassmann convexity JO - Comptes Rendus. Mathématique PY - 2023 SP - 445 EP - 451 VL - 361 IS - G2 PB - Académie des sciences, Paris UR - https://numdam.org/articles/10.5802/crmath.343/ DO - 10.5802/crmath.343 LA - en ID - CRMATH_2023__361_G2_445_0 ER -
%0 Journal Article %A Saldanha, Nicolau %A Shapiro, Boris %A Shapiro, Michael %T Finiteness of rank for Grassmann convexity %J Comptes Rendus. Mathématique %D 2023 %P 445-451 %V 361 %N G2 %I Académie des sciences, Paris %U https://numdam.org/articles/10.5802/crmath.343/ %R 10.5802/crmath.343 %G en %F CRMATH_2023__361_G2_445_0
Saldanha, Nicolau; Shapiro, Boris; Shapiro, Michael. Finiteness of rank for Grassmann convexity. Comptes Rendus. Mathématique, Tome 361 (2023) no. G2, pp. 445-451. doi: 10.5802/crmath.343
[1] Parametrizations of Canonical Bases and Totally Positive Matrices, Adv. Math., Volume 122 (1996) no. 1, pp. 49-149 | DOI | Zbl
[2] Disconjugacy, Lecture Notes in Mathematics, 220, Springer, 1971 | DOI | Zbl
[3] Combinatorialization of spaces of nondegenerate spherical curves (2018) (https://arxiv.org/abs/1810.08632)
[4] Stratification by itineraries of spaces of locally convex curves (2019) (https://arxiv.org/abs/1907.01659v1)
[5] 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] On disconjugate differential equations, Trans. Am. Math. Soc., Volume 134 (1968) no. 1, pp. 53-70 | DOI | Zbl
[7] Grassmann convexity and multiplicative Sturm theory, revisited, Mosc. Math. J., Volume 21 (2021) no. 3, pp. 613-637 | DOI | Zbl
[8] Projective convexity in implies Grassmann convexity, Int. J. Math., Volume 11 (2000) no. 4, pp. 579-588 | DOI | Zbl
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