Disconjugacy and the Secant Conjecture

Arnold Mathematical Journal, Aug 2015

We discuss the so-called secant conjecture in real algebraic geometry, and show that it follows from another interesting conjecture, about disconjugacy of vector spaces of real polynomials in one variable.

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Disconjugacy and the Secant Conjecture

Arnold Math J. Disconjugacy and the Secant Conjecture Alexandre Eremenko 0 Mathematics Subject Classification 0 B Alexandre Eremenko 0 0 Department of Mathematics, Purdue University , West Lafayette, IN 47907 , USA We discuss the so-called secant conjecture in real algebraic geometry, and show that it follows from another interesting conjecture, about disconjugacy of vector spaces of real polynomials in one variable. (a) Every f ∈ V \{0} has at most n − 1 zeros, or (b) For every n distinct points x1, . . . , xn on [a, b] and every basis f1, . . . , fn of V we have det( fi (x j )) = 0. One can replace “every basis” by “some basis” in (b) and obtain an equivalent condition. If V is disconjugate then the determinant in (b) has constant sign which depends only on the ordering of x j and on the choice of the basis. A space of real functions on an open interval is called disconjugate if it is disconjugate on every closed subinterval. Supported by NSF grant DMS-1361836. Disconjugacy; Wronskian; Schubert calculus - 14P99 · 34C10 · 26C10 Let V be a real vector space of dimension n whose elements are real functions on an interval [a, b]. The space V is called disconjugate if one of the following equivalent conditions is satisfied: We are only interested here in spaces V consisting of polynomials. Suppose that a positive integer d is given, and V consists of polynomials of degree a most d. Then every basis f1, . . . , fn of V defines a real rational curve RP1 → RPn−1 of degree d. Indeed, we can replace every f j (x ) by a homogeneous polynomial f j∗(x0, x1) of two variables of degree d, such that f j (x ) = f j∗(1, x ), and then f1∗, . . . , fn∗ define a map f : RP1 → RPn−1 (if polynomials have a common root, divide it out). Then the geometric interpretation of disconjugacy is: (c) The curve f constructed from a basis in V is convex, that is intersects every hyperplane at most n − 1 times. For every basis f1, . . . , fn in V we can consider its Wronski determinant W = W ( f1, . . . , fn ). Changing the basis results in multiplication of W by a non-zero constant, so the roots of W only depend on V . Conjecture 1 Suppose that all roots of W are real. Then V is disconjugate on every interval that does not contain these roots. This is known for n = 2 with arbitrary d (see below), and for n = 3, d ≤ 5 by direct verification with a computer. This conjecture arises in real enumerative geometry (Schubert calculus), and we explain the connection. The problem of enumerative geometry we are interested in is the following: Let m ≥ 2 and p ≥ 2 be given integers. Suppose that mp linear subspaces of dimension p in general position in Cm+ p are given. How many linear subspaces of dimension m intersect all of them non-trivially? The answer was obtained by Schubert in 1886 and it is 1!2! . . . ( p − 1)!(mp)! d(m, p) = m!(m + 1)! . . . (m + p − 1)! . Now suppose that all those given subspaces are real. Does it follow that all p-subspaces that intersect all of them non-trivially are real? The answer is negative, and we are interested in finding a geometric condition on the given p-subspaces that ensure that all d(m, p) subspaces of dimension m that intersect all the given p-subspaces nontrivially are real. One such condition was proposed by B. and M. Shapiro. Let F (x ) = (1, x , . . . , x d ), d = m + p − 1 be a rational normal curve, a. k. a. moment curve. Suppose that the given p-spaces are osculating F at some real points F (x j ). This means that subspaces X j are spanned by the (row)-vectors F (x j ), F (x j ), . . . , F ( p−1)(x j ) for some real x j , 1 ≤ j ≤ mp. Then all m-subspaces that intersect all X j non-trivially are real. This was conjectured by B. and M. Shapiro and proved by Mukhin, Tarasov and Varchenko (MTV) Mukhin et al. (2009) . Earlier it was known for n = 2 Eremenko and Gabrielov (2002), and in Eremenko and Gabrielov (2011) a simplified elementary proof for the case n = 2 was given. We are interested in the following generalization of this result. Secant Conjecture. Suppose that each of the mp subspaces X j , 1 ≤ j ≤ mp is spanned by p distinct real vectors F (x j,k ), 0 ≤ k ≤ p − 1, and that the sets of points {x j,k : 0 ≤ k ≤ p − 1} are separated, that is x j,k ∈ I j , where I j ⊂ RP1 are disjoint intervals. Then all m-subspaces which intersect all X j non-trivially are real. This is known when p = 2, Eremenko et al. (2006) and has been tested on a computer for p = 3 and small m, Hillar et al. (2010) , Garcia-Puente et al. (2012) . The special case when the groups {x j,k }kp=−01 form arithmetic progressions, x j,k = x j,0 + kh has been established Mukhin et al. (2009) . Next we show how the Secant Conjecture follows from Conjecture 1 and the results of MTV. Let us represent an m-subspace Y that intersects all subspaces X j as the zero set of p linear forms, and use the coefficients of these forms as coefficients of p polynomials f0, . . . , f p−1. Then the condition that Y inte (...truncated)


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Alexandre Eremenko. Disconjugacy and the Secant Conjecture, Arnold Mathematical Journal, 2015, pp. 339-342, Volume 1, Issue 3, DOI: 10.1007/s40598-015-0023-5