Disconjugacy and the Secant Conjecture
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.
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:
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(a)
Every $f\in V\backslash\{0\}$ has at most $n-1$ zeros, or
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(b)
For every $n$ distinct points $x_{1},\ldots,x_{n}$ on $[a,b]$ and every basis $f_{1},\ldots,f_{n}$ of $V$ we have $\det(f_{i}(x_{j}))\neq 0$.
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.
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 $f_{1},\ldots,f_{n}$ of $V$ defines a real rational curve $\mathbf{RP}^{1}\to\mathbf{RP}^{n-1}$ of degree $d$. Indeed, we can replace every $f_{j}(x)$ by a homogeneous polynomial $f^{*}_{j}(x_{0},x_{1})$ of two variables of degree $d$, such that $f_{j}(x)=f^{*}_{j}(1,x)$, and then $f^{*}_{1},\ldots,f^{*}_{n}$ define a map $f:\mathbf{RP}^{1}\to\mathbf{RP}^{n-1}$ (if polynomials have a common root, divide it out).
Then the geometric interpretation of disconjugacy is:
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(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 $f_{1},\ldots,f_{n}$ in $V$ we can consider its Wronski determinant $W=W(f_{1},\ldots,f_{n})$. 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 $n=3,d\leq 5$ 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 $m\geq 2$ and $p\geq 2$ be given integers. Suppose that $mp$ linear subspaces of dimension $p$ in general position in $\mathbf{C}^{m+p}$ 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
| $\displaystyle d(m,p)=\frac{1!2!\ldots(p-1)!(mp)!}{m!(m+1)!\ldots(m+p-1)!}.$ |
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 non-trivially are real.
One such condition was proposed by B. and M. Shapiro. Let $F(x)=(1,x,\ldots,x^{d}),{}d=m+p-1$ 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 $F(x_{j}),F^{\prime}(x_{j}),\ldots,F^{(p-1)}(x_{j})$ for some real $x_{j},1\leq j\leq mp$. 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 Gabrielov2002], and in [Eremenko and Gabrielov2011] 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 $X_{j},1\leq j\leq mp$ is spanned by $p$ distinct real vectors $F(x_{j,k}),0\leq k\leq p-1$, and that the sets of points $\{x_{j,k}:0\leq k\leq p-1\}$ are separated, that is $x_{j,k}\in I_{j}$, where $I_{j}\subset\mathbf{RP}^{1}$ 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 $\{x_{j,k}\}_{k=0}^{p-1}$ 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 $f_{0},\ldots,f_{p-1}$. Then the condition that $Y$ intersects some $X_{j}$ is equivalent to linear dependence of the $p$ vectors $f_{i}(x_{j,m})_{m=0}^{p-1},\;i=0,\ldots,p-1$. That is
| $\displaystyle\det(f_{i}(x_{j,m}))_{i,m=0}^{p-1}=0.$ |
These equations for $j=1,\ldots,mp$ define the subspaces $Y$, and we have to prove that all solutions are real.
Let $I_{j}$ be the intervals with disjoint closures which contain the $x_{j,k}$. We may assume without loss of generality that $\infty\notin I_{j}$. We place on each $I_{j}$ a point $y_{j}$, and consider the $d(m,p)$ real rational curves $\mathbf R\to\mathbf{RP}^{p}$ with inflection points at $y_{j}$. These curves exist by the MTV theorem, and they depend continuously on the $y_{j}$.
Let $f=(f_{0}\ldots,f_{p-1})$ be one of these curves. Fix $k\in\{1,\ldots,mp\}$. For all $j\neq k$, fix all $y_{j}\in I_{j}$. When $y_{k}$ moves in $I_{k}$ from the left end to the right end, the determinant $\det(f_{i}(x_{k,m}))_{i,m=0}^{p-1}$ must change sign, in view of Conjecture 1. So this determinant is 0 for some position of $y_{k}$ on $I_{k}$.
Then it follows by a well-known topological argument that one can choose all $y_{j}\in I_{j}$ in such a way that $\det(f_{i}(x_{j,m}))=0$ for all $j$.
Thus we have constructed $d(m,p)$ real solutions of the secant problem. As the total number of solutions is also $d(m,p)$, for generic data, we obtain the result.
Proof of Conjecture 1 for $n=2$. We have two real polynomials $f_{1}$ and $f_{2}$, such that $f_{1}^{\prime}f_{2}-f_{1}f_{2}^{\prime}$ has only real zeros. This means that the rational function $F=f_{1}/f_{2}:\mathbf{\overline{C}}\to\mathbf{\overline{C}}$ is real and all its critical points are real. Let $I\subset\mathbf R$ be a closed interval without critical points. Then $F$ is a local homeomorphism on $I$, so $F(I+i\epsilon)$ belongs to one of the half-planes $\mathbf{C}\backslash\mathbf R$, for all sufficiently small $\epsilon>0$. Suppose without loss of generality that it belongs to the upper half-plane $H$. Let $D$ be the component of $F^{-1}(H)$ that contains $I+i\epsilon$. Then $D$ is a region in $H$ with piecewise analytic boundary, and $I\subset\partial D$. The map $F:D\to H$ is a covering because it is proper and has no critical points. As $H$ is simply connected, $D$ must be simply connected and $F:D\to H$ must be a conformal homeomorphism. Then $F^{-1}:H\to D$ is a conformal homeomorphism. As $\partial D$ is locally connected, this homeomorphism extends to $F^{-1}:\overline{H}\to\overline{D}$. This last map must be injective because this is a left inverse of a function. Thus $F^{-1}:\overline{H}\to\overline{D}$ is a homeomorphism. Then $F:\overline{D}\to\overline{H}$ must be also a homeomorphism, in particular $F$ is injective on $I$.
This implies that the linear span of $f_{1},f_{2}$ is disconjugate.
