Grassmann convexity and multiplicative Sturm theory, revisited

Abstract

In this paper we settle a special case of the Grassmann convexity conjecture formulated in [11]. We present a conjectural formula for the maximal total number of real zeros of the consecutive Wronskians of an arbitrary fundamental solution to a disconjugate linear ordinary differential equation with real time, comp. [13]. We show that this formula gives the lower bound for the required total number of real zeros for equations of an arbitrary order and, using our results on the Grassmann convexity, we prove that the aforementioned formula is correct for equations of orders 4 and 5.Conjecture 1.2 (Upper bound on the number of real zeros of a Wronskian). Given any equation (1.1) disconjugate on I, a positive integer 1 ≤ k ≤ n − 1, and an arbitrary k-tuple (y 1 (x), y 2 (x), . . . , y k (x)) of its linearly independent solutions,

Locally convex curves and the Bruhat stratification of the spin group

“…Conjecture 2.4 in [35] can be regarded as an attempt at a multiplicative Sturm theory for linear differential ODEs of order n+1 > 2; the case n = 1 corresponding to the classical (additive) Sturm theory. The conjecture was proved for n = 2 in [34] and recently for n ≤ 4 in [32], using some material from the present paper, particularly Theorem 4. The said material was also recently applied (in work in progress with E. Alves, B. Shapiro and M. Shapiro) to the problem of counting and classifying connected components of the sets Q −1 [Bru σ ] ⊆ Lo 1 n+1 (for σ ∈ S n+1 ); Theorem 1 and Corollary 6.5 are particularly relevant.…”

“…Section 7 contains the proof of Theorem 4. Section 8 mentions applications of the results of the present paper in [15,32] and work in progress. This paper contains follow-up material inspired by the Ph.…”

Stratification by itineraries of spaces of locally convex curves

“…Conjectures 2.4 and 2.6 of [31] (mentioned earlier in this introduction) are related to an attempt at a generalized (multiplicative) Sturm Theory for linear ordinary differential equations of order n + 1 > 2, the case n = 1 standing for the classical (additive) one. The first of these conjectures has been proved for n ≤ 3 in [27,30], but the general case remains open; the second one is essentially our Lemma 4.1. The second author was first led to consider this subject while studying the critical sets of nonlinear differential operators with periodic coefficients, in a series of works with D. Burghelea and C. Tomei [6,7,8,28].…”