# Real Solutions Of A Problem In Enumerative Geometry

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### Real enumerative geometry and effective algebraic equivalence

solutions to such problems; in particular we ask when a given problem of enumerative geometry can have all its solutions real. We call such a problem filly real. Little is known about enumerative geometry from this perspective. Since the Bezout bound may be attained for real polynomials, the problem of intersecting hypersurfaces in projective

### Rational Curves on Grassmanians: systems theory, reality, and

Abstract. We discuss a particular problem of enumerating rational curves on a Grassman-nian from several perspectives, including systems theory, real enumerative geometry, and symbolic computation. We also present a new transversality result, showing this problem is enumerative in all characteristics.

### Schubert induction - Annals of Mathematics Annals of

1.1. The answer to this problem over C is the prototype of the pro-gram in enumerative geometry. By the Kleiman-Bertini theorem [Kl1], the Schubert conditions intersect transversely, i.e. at a ﬁnite number of reduced points. Hence the problem is reduced to one about the intersection theory of the Grassmannian.

### REALITY AND COMPUTATION IN SCHUBERT CALCULUS A Dissertation

proved this congruence, giving a new invariant in enumerative real algebraic geometry. We also discovered a family of Schubert problems whose number of real solutions to a real osculating instance has a lower bound depending only on the number of de ning ags with real osculation points.

### A probabilistic approach to real enumerative geometry

A probabilistic approach to real enumerative geometry. Antonio Lerario, SISSA Enumerative geometry deals with the problem of counting ( enumerating ) geometric objects satisfying some constraint on their arrangement. For example: how many lines in three-space intersect at the same time four given lines?

### GROMOV{WITTEN THEORY: FROM CURVE COUNTS TO STRING THEORY

enumerative geometry could nally be stated in a robust and rigor-ous way. Still, the actual computation of numerical solutions to those problems remained, in many cases, an unwieldy (if at least well-de ned) task. Another breakthrough was needed in order to open the oodgates to such computations, and in this case, the inspiration came from a

### From Enumerative Geometry to Solving Systems of Polynomial

From Enumerative Geometry to Solving Systems of Polynomial Equations Frank Sottile? Solving a system of polynomial equations is a ubiquitous problem in the applications of mathematics. Until recently, it has been hopeless to nd ex-plicit solutions to such systems, and mathematics has instead developed deep

### Enumerative Algebraic Geometry of Conics

made this endeavor the subject of his Þfteenth challenge problem. Enumerative problems have a long history: many such problems were posed by the ancient Greeks. Enumerative geometry is also currently one of the most active areas of research in algebraic geometry, mainly due to a recent inßux of ideas from string theory.

### How to Count Curves: From Nineteenth-Century Problems to

Keywords: Frobenius manifolds; integrable systems; enumerative geometry 1. Introduction Guess the next term in the sequence 1, 1, 12, 620, 87 304, 26 312 976. This problem belongs to an area of mathematics known as enumerative geometry, the origins of which date from the nineteenth century, when much progress was made,

### Numerical real algebraic geometry

groups of enumerative geometry problems were studied by Jordan in 1870 [Jor70], Harris laid their modern foundations in 1979 [Har79], showing that the algebraic Galois group equals a geometric monodromy group, and computing Galois groups of several such problems.

### Contemporary Schubert Calculus and Schubert Geometry

Contemporary Schubert Calculus and Schubert Geometry Frank Sottile (Texas A& M) Jim Carrell (U.B.C.) March 17-21, 2007 1 A Brief Overview of Schubert Calculus and Related Theories Schubert calculus refers to the calculus of enumerative geometry, which is the art of counting geometric ﬁgures determined by given incidence conditions.

### oaktrust.library.tamu.edu

ABSTRACT The Mukhin-Tarasov-Varchenko Theorem (previously the Shapiro Conjecture) as-serts that a Schubert problem has all solutions distinct and real if the Schubert varieties in

### Real solutions to systems of polynomial equations and

the following real enumerative geometric theorem. Theorem 1. There exists eight lines in R3 met by 92 real plane conics. Since eight general lines in C3 are met by 92 plane conics, Theorem 1 shows that the real analog of this enumerative geometry problem is answered in the a rmative. 1 Background 1.1 Algebraic sets For a polynomial system g: CN

### Enumerative geometry - UCSD Mathematics Home

Principle 9.2. (Principle of continuity) If we are given a problem in enumerative geometry, then the number of solutions is invariant under a continuous change of parameters. This is a very useful principle; unfortunately as stated it is clearly false, as there are some obvious counterexamples. The point is to

### Problem Session for Numerical Algebraic Geometry

A classical enumerative geometry problem is to count the number of different choices of 6 real lines to count the possible number of real solutions. (Open) d

### Rational functions with real critical points and the B. and M

Theorems A and 2 imply Theorem 1. In general, even if the lines in Problem P are real, the subspaces of codimension two might not be real . Fulton  asked the following general question: how many solutions of a problem of enumerative geometry can be real, when that problem is one of counting

### REAL SCHUBERT CALCULUS: POLYNOMIAL SYSTEMS AND A CONJECTURE

Determining the number of real solutions to a system of polynomial equations is a chal-lenging problem in symbolic and numeric computation [19, 48, 49] with real world appli-cations . Related questions include when a problem of enumerative geometry can have all solutions real  and when may a given physical system be controlled by real

### 3264ConicsinaSecond

15th problem and thus to the twentieth-century develop-ment of enumerative algebraic geometry. The number 3264appears prominently in the title of the textbook by Eisenbud and Harris [EH16]. A delightful introduction toSteiner sproblemwaspresentedbyBashelor,Ksir,and Travesin[BKT08]. Numerical algebraic geometry is a younger subject. It

### Linear ordinary differential equations and Schubert calculus

the qualitative theory of linear ordinary diﬀerential equations with real time and the reality problems in Schubert calculus. We formulate a few relevant conjectures. 1. Introduction Questions asking under what conditions a given enumerative problem in geometry with all real initial data has all real solutions have a long history and appear often

### Cylinders Through Five Points: Computational Algebra and Geometry

Key words and phrases: Computational geometry, enumerative geometry, Gröbner bases, nonlinear systems, symbolic−numeric computation. 1. Outline of the Problem and Related Work Given five points in R3, we are to determine all right circular cylinders containing those points. We do this by solving equations for the axial line and radius

### Introduction

CENTURY SOLUTIONS IAN STRACHAN 1. Introduction Guess the next term in the sequence: 1, 1, 12, 620, 87304, 26312976. This problem belongs to an area of mathematics known as enumerative geometry, the origins of which date from the 19th century, when much progress was made, and even earlier to classical Euclidean geometry.

### Some Real and Unreal Enumerative Geometry for Flag Manifolds

Fulton  asked how many solutions to such a problem of enumerative geometry can be real and later with Pragacz  reiterated this question in the context of ﬂag manifolds. It is interesting that in every known case, all solutions may be real. These in-clude the classical problem of 3264 plane conics tangent to 5 plane conics , the

### QUALITATIVE ASPECTS OF COUNTING REAL RATIONAL CURVES ON REAL

count of real rational curves interpolating real collections of points on a real ra-tional surface has allowed to respond in an afﬁrmative way to the long standing problem of existence of real solutions in this enumerative problem. Moreover, the lower bound on the number of real solutions provided by the Welschinger invari-

### Rational Functions with Real Critical Points and the B. and M

(see [8, p. 55]): how many solutions of real equations can be real, particularly for enumerative problems? We refer to a recent survey  of results related to this question. A specific conjecture for the Problem P was made by Boris and Michael Shapiro (see, for example, ): if the lines in question are tangent to the rational normal curve

### Cylinders Through Five Points: Complex and Real Enumerative

which we go from six to zero real solutions. Also of interest: it is closely related to a case where the common tangent to four given spheres has all solutions real (12 of them). I have a vague belief, which I elevate to conjecture , that any config− uration with six solutions is in some sense a perturbation of one of the two described above.

### Problem Session for Numerical Algebraic Geometry

the maximum number of real solutions as a function of r= rank Band n. 3 3. b.A classical enumerative geometry problem is to count the number of plane conics in C3

### Problem-Solving Strategies - MATHEMATICAL OLYMPIADS

R the real numbers R+ the positive real numbers C the complex numbers Z n the integers modulo n 1 nthe integers 1, 2, ,n Notations from Sets, Logic, and Geometry ⇐⇒ iff, if and only if ⇒ implies A ⊂BAis a subset of B A BAwithout B A∩B the intersection of A and B A∪B the union of A and B a ∈A the element a belongs to the set A

### RANDOM FIELDS AND THE ENUMERATIVE GEOMETRY OF LINES ON REAL

number of solutions is not de ned (e.g. the signed count or the average count). Our contribution with this paper is thus in two di erent directions. On one hand we present new results in the emerging eld of random real algebraic geometry, with the investigation of the real average count.

### Rational functions with real critical points and the B. and M

(see [8, p. 55]): how many solutions of real equations can be real, particu-larly for enumerative problems? We refer to a recent survey  of results related to this question. A speci c conjecture for the Problem P was made by Boris and Michael Shapiro (see, for example, ): if the lines in question are tangent to the rational normal curve

### Solutions For Geometry By David Brannan

Computational Conformal Geometry Solutions Manual for Chapters 1-10, Calculus with Analytic Geometry Geometry Workbook Instructor's solutions manual Problems and Solutions in Euclidean Geometry This book can form the basis of a second course in algebraic geometry. As motivation, it takes concrete questions from enumerative geometry

### EXPERIMENTATION AND CONJECTURES IN THE REAL SCHUBERT CALCULUS

The Shapiro conjecture for Grassmannians [24, 18] has driven progress in enumerative real algebraic geometry , which is the study of real solutions to geometric problems. It conjectures that a (zero-dimensional) intersection of Schubert subvarieties of a Grass-

### Towards enumerative theories for structures on 4-manifolds

Fredholm topology and enumerative geometry: reflections on some words of Michael Atiyah To appear in Proceedings of the Gokova conference on geometry and topology, special issue in memory of Atiyah. The paper and talk are to some degree speculative. Simon Donaldson Towards enumerative theories for structures on 4-manifolds

### Applications and Combinatorics in Algebraic Geometry

braic Geometry: Real Solutions, Applications, and Combinatorics (September 2007 August 2010). Another NSF grant, Numerical Real Algebraic Geometry DMS-0915211, has just started. This primarily supports a postdoc (Jon Hauenstein), a graduate student (Abra-ham Mart´ın del Campo), and an undergraduate student (Christopher Brooks) to

### First Joint Meeting between the RSME and the AMS

On the enumerative geometry of real algebraic curves Johan Huisman (Universit´e Rennes) Let C be a smooth real plane curve. Let c be its degree and g its genus. We assume that C has at least g real branches, i.e., C is either an M-curve or an (M − 1)-curve. Let d be a nonzero natural integer strictly less than c. Let e be a partition of cd

### Mittag-Leffler Institute

Contents 1 Overview 1 1.1 Upper bounds 3 1.2 The Wronski map and the Shapiro Conjecture

### Board on Mathematical Sciences & Analytics

algebraic geometry. As motivation, it takes concrete questions from enumerative geometry and intersection theory, and provides intuition and technique, so that the student develops the ability to solve geometric problems. The authors explain key ideas, including rational equivalence, Chow rings, Schubert calculus and Chern classes, and readers

### Enumerative Algebraic Geometry of Conics

enumerative algebraic geometry on a ﬁrm mathematical foundation. Indeed, Hilbert made this endeavor the subject of his ﬁfteenth challenge problem. Enumerative problems have a long history: many such problems were posed by the ancient Greeks. Enumerative geometry is also currently one of the most active

### arXiv:math/0007142v1 [math.AG] 24 Jul 2000

geometry, determining the number of solutions is the central problem in enumerative geometry. Example 1.1.1. We illustrate this last paragraph with an example. Let f1,f2,f3,f4 be random quadratic polynomials in the ring F101[y11,y12,y21,y22]. i1 : R=ZZ/101[y11, y12, y21, y22]; Date: March 10, 2008.

### Cylinders through Five Points: Computational Algebra and Geometry

The problem of ﬁnding cylinders through ﬁve points may be recast in a computational geometry setting: Given ﬁve points in R 3 , ﬁnd the smallest positive r , and corresponding orientation