Alexandre Eremenko, Andrei Gabrielov
We give a new elementary proof of the following theorem: if all critical points of a rational function g belong to the real line then there exists a fractional linear transformation L such that L(g) is a real rational function. Then we interpret the result in terms of Fuchsian differential equations whose general solution is a polynomial and in terms of electrostatics.
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Suppose that 2d-2 tangent lines to the rational normal curve z\mapsto (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…
Motivated by the Shapiro Shapiro conjecture, we consider the following: given a field $k$, under what conditions must a rational function with only $k$-rational ramification points be equivalent (after post-composition with a fractional linear transformation) to a rational function defined over $k$? The main results of this paper answer this question when $k$ has characteristic 2 or 3. We also show the insufficiency of several natural conditions in higher characteristic.
We prove a criterion for the reality of irreducible representations of the rational Cherednik algebras H_{0,1}(S_n). This is shown to imply a criterion for the real loci of the Calogero-Moser spaces C_n in terms of the Etingof-Ginzburg finite maps Υ\colon C_n \to C^n/S_n \times C^n/S_n, recovering a result of Mikhin, Tarasov, and Varchenko [MTV2]. As a consequence we obtain a criterion for the real locus of the Wilson's adelic Grassmannian of rank one bispectral solutions of the KP hierarchy. Using Wilson's first parametrisation of the adelic Grassmannian, we give a new proof of a result of [M…
This paper has been withdrawn by the authors; it is superceeded by the paper: The real loci of Calogero--Moser spaces, representations of rational Cherednik algebras and the Shapiro conjecture, by Iain Gordon, Emil Horozov, and Milen Yakimov, arXiv:0711.4336.
The purpose of this note is to give a refinement of the product formula proved in [1] for the Poincare polynomial of a smooth Schubert variety in the flag variety of an algebraic group G over C. This yields a factorization of the number of elements in a Bruhat interval [e,w] in the Weyl group W of G provided the Schubert variety associated to w is smooth. This gives an elementary necessary condition for a Schubert variety in the flag variety to be smooth.
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.
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 infle…
We prove that the moduli spaces of K3 surfaces with non-symplectic involution are rational for four deformation types. With the previous results, this establishes the rationality of those moduli spaces except two classical cases.
We prove an elementary method to classify, up to rigid isotopy, all nodal degree 4 real rational curves in $\mathbb{RP}^2$ that have only real double points. We show how to associate a chord diagram to a nodal real degree 4 planar rational curve and prove that such curves are defined up to rigid isotopy by their chord diagrams.
2--elementary rational covers of the plane
本文研究了平面中Galois群形式为$\mathbb Z_2^r$的初等有理覆盖的分类问题。
We prove the B. and M. Shapiro conjecture that says 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 $ f : CP^1 \to CP^r $ lie on a circle in the Riemann sphere $ CP^1 $, then $f$ maps this circle into a suitable real subspace $ RP^r \subset CP^r $. The proof is based on the Bethe ansatz method in the Gaudin model. The key observation is that a symmetric linea…
We study the Hilbert series of a family of ideals J_ϕgenerated by powers of linear forms in k[x_1,...,x_n]. Using the results of Emsalem-Iarrobino, we formulate this as a question about fatpoints in P^{n-1}. In the three variable case this is equivalent to studying the dimension of a linear system on a blow up of P^2. The ideals that arise have the points in very special position, but because there are only seven points, we can apply results of Harbourne to obtain the classes of the negative curves. Reducing to an effective, nef divisor and using Riemann-Roch yields a formula for the Hilbert s…
Torsion of elliptic curves with rational $j$-invariant over the maximal elementary abelian 2-extension of $\mathbb{Q}$
本文针对定义在有理数域所有二次扩张合成域上、j不变量为非0或1728的有理数的椭圆曲线，完成了对其可能挠子群结构的分类。
We study the birational geometry of a Fano 4-fold X from the point of view of Mori dream spaces; more precisely, we study rational contractions of X. Here a rational contraction is a rational map f: X-->Y, where Y is normal and projective, which factors as a finite sequence of flips, followed by a surjective morphism with connected fibers. Such f is called elementary if the difference of the Picard numbers of X and Y is 1. We first give a characterization of non-movable prime divisors in X, when X has Picard number at least 6; this is related to the study of birational and divisorial elementar…
We give a sharp bound for orders of elementary abelian 2-groups of birational automorphisms of rationally connected threefolds.
In this paper, we continue our study of abstract representations of elementary subgroups of Chevalley groups of rank $\geq 2.$ First, we extend our earlier methods to analyze representations of elementary groups over arbitrary associative rings, and as a consequence, prove the conjecture of Borel and Tits on abstract homomorphisms of the groups of rational points of algebraic groups for groups of the form ${\bf SL}_{n,D}$, where $D$ is a finite-dimensional central division algebra over a field of characteristic zero. Second, we apply our results to study deformations of representations of elem…
Let V be a finite dimensional vector space over a local field. Let us say that a complex function on V is elementary if it is a product of the additive character of a rational function Q on V and multiplicative characters of polynomials on V. In this paper we study when the Fourier transform of an elementary function is elementary. If Q has a nonzero Hessian, a necessary condition for this is that the Legendre transform Q_* of Q is rational. The basic example is a nondegenerate quadratic form. We study such functions Q, give examples, and find all of them such that both Q and Q_* are of the fo…
In view of Mori theory, rational homogenous manifolds satisfy a recursive condition: every elementary contraction is a rational homogeneous fibration and the image of any elementary contraction also satisfies the same property. In this paper, we show that a smooth Fano $n$-fold with the same condition and Picard number greater than $n-6$ is either a rational homogeneous manifold or the product of $n-7$ copies of $\mathbb{P}^1$ and a Fano $7$-fold $X_0$ constructed by G. Ottaviani. We also clarify that $X_0$ has non-nef tangent bundle and in particular is not rational homogeneous.
In this short note we give an elementary combinatorial argument, showing that the Conjecture of J. Fernández de Bobadilla, I. Luengo, A. Melle-Hernández, A. Némethi follows from the results of M. Borodzik and C. Livingston in the case of rational cuspidal curves with two critical points.
We exploit an elementary specialization technique to study some properties of rational curves on index $n-1$ Fano $n$-folds. We prove a simple formula for counting rational curves passing through a suitable number of points in the case $n=3$. The arguments have immediate deformation theoretic consequences which translate to properties of the normal bundles of such curves.
Points of bounded height on quintic del Pezzo surfaces over the rational numbers
本文针对有理数域上分裂光滑五次del Pezzo曲面，给出了Manin猜想的一个相对简短且初等的证明。
In this paper, we give a short proof of a theorem by Kollár on hereditarily rational functions. This is an answer to his appeal to find an elementary proof which does not rely so much on resolution of singularities. Our approach does not make use of desingularization techniques. Instead, we apply a stronger version of the Łojasiewicz inequality. Moreover, this allows us to sharpen Kollár's theorem.
This note studies, and partially solves, 3 elementary questions about continuous rational functions on real (and p-adic) algebraic varieties: Can one restrict such a function to a subvariety? Can one extend such a function from a subvariety? Which linear equations can be solved using such functions?
In this note we deal with rational curves in P^3 which are images of a line by means of a finite sequence of cubo-cubic Cremona transformations. We prove that these curves can always be obtained applying to the line a sequence of such transformations increasing at each step the degree of the curve. As a corollary we get a result about curves that can give speciality for linear systems of P^3.
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 it as well as computational evidence obtained in over one terahertz-year of computing, and we discuss some phenomena we observed in our data.
This note replaces two earlier preprints (1101.3737 by Kollár) and (1211.6681 by Nowak). It studies, and partially solves, 3 elementary questions about continuous rational functions on real (and p-adic) algebraic varieties: Can one restrict such a function to a subvariety? Can one extend such a function from a subvariety? Which linear equations can be solved using such functions? v.2: Section 3 simplified and typos corrected.
The periods, introduced by Kontsevich and Zagier, form a class of complex numbers which contains all algebraic numbers and several transcendental quantities. Little has been known about qualitative properties of periods. In this paper, we compare the periods with hierarchy of real numbers induced from computational complexities. In particular we prove that periods can be effectively approximated by elementary rational Cauchy sequences. As an application, we exhibit a computable real number which is not a period.
A complex projective manifold is rationally connected, resp. rationally simply connected, if finite subsets are connected by a rational curve, resp. the spaces parameterizing these connecting rational curves are themselves rationally connected. We prove that a projective scheme over a global function field with vanishing "elementary obstruction" has a rational point if it deforms to a rationally simply connected variety in characteristic 0. This gives new, uniform proofs over these fields of the Period-Index Theorem, the quasi-split case of Serre's "Conjecture II", and Lang's $C_2$ property.
Over an algebraically closed field of positive characteristic, there exist rational functions with only one critical point. We give an elementary characterization of these functions in terms of their continued fraction expansions. Then we use this tool to discern some of the basic geometry of the space of unicritical rational functions, as well as its quotients by the SL(2)-actions of conjugation and postcomposition. We also give an application to dynamical systems with restricted ramification defined over non-Archimedean fields of positive residue characteristic.
We consider the structure ${\mathbb R}^{\mathrm{RE}}$ obtained from $({\mathbb R},<,+,\cdot)$ by adjoining the restricted exponential and sine functions. We prove Wilkie's conjecture for sets definable in this structure: the number of rational points of height $H$ in the transcendental part of any definable set is bounded by a polynomial in $\log H$. We also prove two refined conjectures due to Pila concerning the density of algebraic points from a fixed number field, or with a fixed algebraic degree, for ${\mathbb R}^{\mathrm{RE}}$-definable sets.
Enumerating ramified coverings of the sphere with fixed ramification types is a well-known problem first considered by A. Hurwitz. Up to now, explicit solutions have been obtained only for some families of ramified coverings, for instant, those realized by polynomials in one complex variable. In this paper we obtain an explicit answer for a large new family of coverings, namely, the coverings realized by simple almost polynomials, defined below. Unlike most other results in the field, our formula is obtained by elementary methods.
This short expository note gives an elementary introduction to the study of dynamics on certain moduli spaces, and in particular the recent breakthrough result of Eskin, Mirzakhani, and Mohammadi. We also discuss the context and applications of this result, and connections to other areas of mathematics such as algebraic geometry, Teichmuller theory, and ergodic theory on homogeneous spaces.
Semidefinite representation of convex hulls of rational varieties
利用正半定矩矩阵和多项式平方和分解的对偶性质，证明了有理参数化代数簇的凸包在曲线、二次参数化超曲面和双变量四次参数化超曲面情形下具有半定表示。
The central objective of this dissertation was to present the Goppa Geometry Codes via elementary methods which were introduced by J.H. van Lint ,R.Pellikaan and T. Hohold about 1998. On the first part of such dissertation are presented the fundamental concepts about bodies of rational functions of an algebraic curve in the direction as to define the Goppa Codes on a classical manner. In this study we based ourselves mainly on the book ? Algebraic Function Fields and Codes? of H. Stichtenoth. The second part is initiated with an introduction about the functions weight, degree and order which a…
The group structure on the rational points of elliptic curves plays several important roles, in mathematics and recently also in other areas such as cryptography. However, the famous proofs for the group property (in particular, for its associative law) require somewhat advanced mathematics and therefore are not easily accessible by non-mathematician. On the other hand, there have been attempts in the literature to give an elementary proof, but those rely on computer-aided calculation for some part in their proofs. In this paper, we give a self-contained proof of the associative law for this o…
We establish several useful commutative diagrams consisting of low term exact sequences attached to {\Grot} spectral sequences, which extends and integrates the previous ones appeared in literature such as Alexei~N. Skorobogatov [Beyond the {M}anin obstruction, Invent. Math. (1999)], and [On the elementary obstruction to the existence of rational points, Mathematical Notes (2007)]. Parts of the diagrams was frequently used in local-global principle to rational points.
We give an elementary introduction to our papers relating the geometry of rational homogeneous varieties to representation theory. We also describe related work and recent progress.
We give a 'recursive' formula (in terms of reducible limits) for counting rational curves on a variety moving in any sufficiently large and well-behaved family. Our approach is completely elementary and makes no use of moduli spaces for maps or the like.
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