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
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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)