Enumerative Real Algebraic Geometry

Enumerative Geometry is concerned with the number of solutions to a structured system of polynomial equations, when the structure comes from geometry. Enumerative real algebraic geometry studies real solutions to such systems, particularly a priori information on their number. Recent results in this area have, often as not, uncovered new and unexpected phenomena, and it is far from clear what to expect in general. Nevertheless, some themes are emerging. This comprehensive article describe the current state of knowledge, indicating these themes, and suggests lines of future research. In particular, it compares the state of knowledge in Enumerative Real Algebraic Geometry with what is known about real solutions to systems of sparse polynomials.Comment: Revised, corrected version. 40 pages, 18 color .eps figures. Expanded web-based version at http://www.math.umass.edu/~sottile/pages/ERAG/index.htm

Real enumerative geometry and effective algebraic equivalence

We describe an approach to the question of finding real solutions to problems of enumerative geometry, in particular the question of whether a problem of enumerative geometry can have all of its solutions be real. We give some methods to infer one problem can have all of its solutions be real, given that a related problem does. These are used to show many Schubert-type enumerative problems on some flag manifolds can have all of their solutions be real.Comment: 12 pages, LaTeX 2

Quantum indices and refined enumeration of real plane curves

We associate a half-integer number, called {\em the quantum index}, to algebraic curves in the real plane satisfying to certain conditions. The area encompassed by the logarithmic image of such curves is equal to $\pi^2$ times the quantum index of the curve and thus has a discrete spectrum of values. We use the quantum index to refine real enumerative geometry in a way consistent with the Block-G\"ottsche invariants from tropical enumerative geometry.Comment: Version 4: exposition improvement, particularly in the proof of Theorem 5 (following referee suggestions

Enumerative geometry for real varieties

We discuss the problem of whether a given problem in enumerative geometry can have all of its solutions be real. In particular, we describe an approach to problems of this type, and show how this can be used to show some enumerative problems involving the Schubert calculus on Grassmannians may have all of their solutions be real. We conclude by describing the work of Fulton and Ronga-Tognoli-Vust, who (independently) showed that there are 5 real plane conics such that each of the 3264 conics tangent to all five are real.Comment: Based upon the Author's talk at 1995 AMS Summer Research Institute in Algebraic geometry. To appear in the Proceedings. 11 pages, extended version with Postscript figures and appendix available at http://www.msri.org/members/bio/sottile.html, or by request from Author ([email protected]

Reelle Algebraische Geometrie

This workshop was organized by Michel Coste (Rennes), Claus Scheiderer (Konstanz) and Niels Schwartz (Passau). The talks focussed on recent developments in real enumerative and tropical geometry, positivity and sums of squares, real aspects of classical algebraic geometry, semialgebraic and tame geometry, and topology and singularities of real varieties

Real Enumerative Questions in Complex and Tropical Geometry

The workshop Real Enumerative Questions in Complex and Tropical Geometry was devoted to a wide discussion and exchange of ideas between the best experts representing various points of view on the subject. Enumeration of real curves largely motivated the development of the tropical geometry and led to the discovery of new interesting geometric phenomena and deep links between this problematic and algebraic geometry, symplectic geometry, topology, and mathematical physics

Intrinsic signs and lower bounds in real algebraic geometry

A classical result due to Segre states that on a real cubic surface in ℙ ℝ 3 $\mathbb {P}^3_\mathbb {R}$ there exist two kinds of real lines: elliptic and hyperbolic lines. These two kinds of real lines are defined in an intrinsic way, i.e., their definition does not depend on any choices of orientation data. Segre's classification of smooth real cubic surfaces also shows that any such surface contains at least 3 real lines. Starting from these remarks and inspired by the classical problem mentioned above, our article has the following goals: (a) We explain a general principle which leads to lower bounds in real algebraic geometry. (b) We explain the reason for the appearance of intrinsic signs in the classical problem treated by Segre, showing that the same phenomenon occurs in a large class of enumerative problems in real algebraic geometry. (c) We illustrate these principles in the enumerative problem for real lines in real hypersurfaces of degree 2m-3$2m-3$ in ℙ ℝ m $\mathbb {P}^m_\mathbb {R}

An excursion from enumerative goemetry to solving systems of polynomial equations with Macaulay 2

Solving a system of polynomial equations is a ubiquitous problem in the applications of mathematics. Until recently, it has been hopeless to find explicit solutions to such systems, and mathematics has instead developed deep and powerful theories about the solutions to polynomial equations. Enumerative Geometry is concerned with counting the number of solutions when the polynomials come from a geometric situation and Intersection Theory gives methods to accomplish the enumeration. We use Macaulay 2 to investigate some problems from enumerative geometry, illustrating some applications of symbolic computation to this important problem of solving systems of polynomial equations. Besides enumerating solutions to the resulting polynomial systems, which include overdetermined, deficient, and improper systems, we address the important question of real solutions to these geometric problems. The text contains evaluated Macaulay 2 code to illuminate the discussion. This is a chapter in the forthcoming book "Computations in Algebraic Geometry with Macaulay 2", edited by D. Eisenbud, D. Grayson, M. Stillman, and B. Sturmfels. While this chapter is largely expository, the results in the last section concerning lines tangent to quadrics are new.Comment: LaTeX 2e, 22 pages, 1 .eps figure. Source file (.tar.gz) includes Macaulay 2 code in article, as well as Macaulay 2 package realroots.m2 Macaulay 2 available at http://www.math.uiuc.edu/Macaulay2 Revised with improved exposition, references updated, Macaulay 2 code rewritten and commente