Shapiro Conjecture Research Articles

An identity in the Bethe subalgebra of C[Sn]$\mathbb {C}[\mathfrak {S}_n

AbstractAs part of the proof of the Bethe ansatz conjecture for the Gaudin model for , Mukhin, Tarasov, and Varchenko described a correspondence between inverse Wronskians of polynomials and eigenspaces of the Gaudin Hamiltonians. Notably, this correspondence afforded the first proof of the Shapiro–Shapiro conjecture. In this paper, we give an identity in the group algebra of the symmetric group, which allows one to establish the correspondence directly, without using the Bethe ansatz.

Large deviations of multichordal $\operatorname{SLE}_{0+}$, real rational functions, and zeta-regularized determinants of Laplacians

We prove a strong large deviation principle (LDP) for multiple chordal \operatorname{SLE}_{0+} curves with respect to the Hausdorff metric. In the single-chord case, this result strengthens an earlier partial result by the second author. We also introduce a Loewner potential, which in the smooth case has a simple expression in terms of zeta-regularized determinants of Laplacians. This potential differs from the LDP rate function by an additive constant depending only on the boundary data, which satisfies PDEs arising as a semiclassical limit of the Belavin–Polyakov–Zamolodchikov equations of level 2 in conformal field theory with central charge c \to -\infty . Furthermore, we show that every multichord minimizing the potential in the upper half-plane for given boundary data is the real locus of a rational function and is unique, thus coinciding with the \kappa \to 0+ limit of the multiple \operatorname{SLE}_\kappa . As a by-product, we provide an analytic proof of the Shapiro conjecture in real enumerative geometry, first proved by Eremenko and Gabrielov: if all critical points of a rational function are real, then the function is real up to post-composition with a Möbius transformation.

A topological proof of the Shapiro–Shapiro conjecture

We prove a generalization of the Shapiro–Shapiro conjecture on Wronskians of polynomials, allowing the Wronskian to have complex conjugate roots. We decompose the real Schubert cell according to the number of real roots of the Wronski map, and define an orientation of each connected component. For each part of this decomposition, we prove that the topological degree of the restricted Wronski map is given as an evaluation of a symmetric group character. In the case where all roots are real, this implies that the restricted Wronski map is a topologically trivial covering map; in particular, this gives a new proof of the Shapiro–Shapiro conjecture.

The Khavinson–Shapiro Conjecture and the Bergman Projection in One and Several Complex Variables

We reveal a complex analogue to a result about polynomial solutions to the Dirichlet Problem on ellipsoids in $$\mathbb {R}^n$$ by showing that the Bergman projection on any ellipsoid in $$\mathbb {C}^n$$ is such that the projection of any polynomial function of degree at most N is a holomorphic polynomial function of degree at most N. The discussion is motivated by a connection between the Bergman projection and the Khavinson–Shapiro conjecture in $$\mathbb {C}$$ . We also relate the Khavinson–Shapiro conjecture to polyharmonic Bergman projections in $$\mathbb {R}^n$$ by showing that these projections take polynomials to polynomials on ellipsoids.

The Monotone Secant Conjecture in the Real Schubert Calculus

The monotone secant conjecture posits a rich class of polynomial systems, all of whose solutions are real. These systems come from the Schubert calculus on flag manifolds, and the monotone secant conjecture is a compelling generalization of the Shapiro conjecture for Grassmannians (theorem of Mukhin, Tarasov, and Varchenko). We present some theoretical evidence for this conjecture, as well as computational evidence obtained by 1.9 terahertz-years of computing, and we discuss some of the phenomena we observed in our data.

The Secant Conjecture in the Real Schubert Calculus

We formulate the secant conjecture, which is a generalization of the Shapiro conjecture for Grassmannians. It asserts that an intersection of Schubert varieties in a Grassmannian is transverse with all points real if the flags defining the Schubert varieties are secant along disjoint intervals of a rational normal curve. We present theoretical evidence for this conjecture as well as computational evidence obtained in over one terahertz-year of computing, and we discuss some of the phenomena we observed in our data.

Some Lower Bounds in the B. and M. Shapiro Conjecture for Flag Varieties

The B. and M. Shapiro conjecture stated that all solutions of the Schubert Calculus problems associated with real points on the rational normal curve should be real. For Grassmannians, it was prove...

Schanuel's Conjecture and Algebraic Roots of Exponential Polynomials

In this article, I will prove that assuming Schanuel's conjecture, an exponential polynomial with algebraic coefficients can have only finitely many algebraic roots. Furthermore, this proof demonstrates that there are no unexpected algebraic roots of any such exponential polynomial. This implies a special case of Shapiro's conjecture: if p(x) and q(x) are two exponential polynomials with algebraic coefficients, each involving only one iteration of the exponential map, and they have common factors only of the form exp (g) for some exponential polynomial g, then p and q have only finitely many common zeros.

Reality Property of Discrete Wronski Map with Imaginary Step

For a set of quasi-exponentials with real exponents, we consider the discrete Wronskian (also known as Casorati determinant) with pure imaginary step 2h. We prove that if the coefficients of the discrete Wronskian are real and for every its roots the imaginary part is at most |h|, then the complex span of this set of quasi-exponentials has a basis consisting of quasi-exponentials with real coefficients. This result is a generalization of the statement of the B. and M. Shapiro conjecture on spaces of polynomials. The proof is based on the Bethe ansatz for the XXX model.

Some Lower Bounds in the B. and M. Shapiro Conjecture for Flag Varieties

The B. and M. Shapiro conjecture stated that all solutions of the Schubert Calculus problems associated with real points on the rational normal curve should be real. For Grassmannians, it was proved by Mukhin, Tarasov, and Varchenko. For flag varieties, Sottile found a counterexample and suggested that all solutions should be real under certain monotonicity conditions. In this paper, we compute lower bounds on the number of real solutions for some special cases of the B. and M. Shapiro conjecture for flag varieties, when Sottile’s monotonicity conditions are not satisfied.

The Khavinson–Shapiro conjecture and polynomial decompositions

The main result of the paper states the following: Let ψ be a polynomial in n variables. Suppose that there exists a constant C>0 such that any polynomial f has a polynomial decomposition f=ψqf+hf with Δkhf=0 and degqf⩽degf+C. Then degψ⩽2k. Here Δk is the kth iterate of the Laplace operator Δ. As an application, new classes of domains in Rn are identified for which the Khavinson–Shapiro conjecture holds.

Frontiers of reality in Schubert calculus

The theorem of Mukhin, Tarasov, and Varchenko (formerly the Shapiro conjecture for Grassmannians) asserts that all (a priori complex) solutions to certain geometric problems in the Schubert calculus are actually real. Their proof is quite remarkable, using ideas from integrable systems, Fuchsian differential equations, and representation theory. There is now a second proof of this result, and it has ramifica- tions in other areas of mathematics, from curves to control theory to combinatorics. Despite this work, the original Shapiro conjecture is not yet settled. While it is false as stated, it has several interesting and not quite understood modifications and gen- eralizations that are likely true, and the strongest and most subtle version of the Shapiro conjecture for Grassmannians remains open.

ON REALITY PROPERTY OF WRONSKI MAPS

We prove that if all roots of the discrete Wronskian with step 1 of a set of quasi-exponentials with real bases are real, simple and differ by at least 1, then the complex span of this set of quasi-exponentials has a basis consisting of quasi-exponentials with real coefficients. This theorem generalizes the statement of the B. and M. Shapiro conjecture about spaces of polynomials. The proof is based on the Bethe ansatz method for the XXX model.

The B. and M. Shapiro conjecture in real algebraic geometry and the Bethe ansatz

We prove the B. and M. Shapiro conjecture that if the Wronskian of a set of polynomials has real roots only, then the complex span of this set of polynomials has a basis consisting of polynomials with real coefficients. This, in particular, implies the following result: If all ramification points of a parametrized rational curve φ: ℂℙ 1 → ℂℙ r lie on a circle in the Riemann sphere ℂℙ 1 , then φ maps this circle into a suitable real subspace ℝℙ r ⊂ ℂℙ r . The proof is based on the Bethe ansatz method in the Gaudin model. The key observation is that a symmetric linear operator on a Euclidean space has real spectrum. In Appendix A, we discuss properties of differential operators associated with Bethe vectors in the Gaudin model. In particular, we prove a statement, which may be useful in complex algebraic geometry; it claims that certain Schubert cycles in a Grassmannian intersect transversally if the spectrum of the corresponding Gaudin Hamiltonians is simple. In Appendix B, we formulate a conjecture on reality of orbits of critical points of master functions and prove this conjecture for master functions associated with Lie algebras of types A r , B r and C r .

Maximally Inflected Real Rational Curves

We introduce and begin the topological study of real rational plane curves, all of whose inflection points are real. The existence of such curves is a corollary of results in the real Schubert calculus, and their study has consequences for the important Shapiro and Shapiro conjecture in the real Schubert calculus. We establish restrictions on the number of real nodes of such curves and construct curves realizing the extreme numbers of real nodes. These constructions imply the existence of real solutions to some problems in the Schubert calculus. We conclude with a discussion of maximally inflected curves of low degree.

Rational Functions with Real Critical Points and the B. and M. Shapiro Conjecture in Real Enumerative Geometry

Suppose that 2d - 2 tangent lines to the rational normal curve z → (1: z …: z d ) in d-dimensional complex projective space are given. It was known that the number of codimension 2 subspaces intersecting all these lines is always finite; for a generic configuration it is equal to the d th Catalan number. We prove that for real tangent lines, all these codimension 2 subspaces are also real, thus confirming a special case of a general conjecture of B. and M. Shapiro. This is equivalent to the following result: If all critical points of a rational function lie on a circle in the Riemann sphere (for example, on the real line), then the function maps this circle into a circle.