Understanding, finding, or even deciding on the existence of real solutions to
a system of equations is a very diﬃcult problem with many applications outside of
mathematics. While it is hopeless to expect much in general, we know a surprising
amount about these questions for systems which possess additional structure coming
from geometry. Such equations from geometry for which we have information about
their real solutions are the subject of this book.
This book focuses on equations from toric varieties and Grassmannians. Not
only is much known in these cases, but they encompass some of the most common
applications. The results may be grouped into three themes:
(I) Upper bounds on the number of real solutions.
(II) Geometric problems that can have all solutions be real.
(III) Lower bounds on the number of real solutions.
Upper bounds (I) bound the complexity of the set of real solutions—they are one of
the sources for the theory of o-minimal structures which are an important topic in
real algebraic geometry. The existence (II) of geometric problems that can have all
solutions be real was initially surprising, but this phenomenon now appears to be
ubiquitous. Lower bounds (III) give existence proofs of real solutions. Their most
spectacular manifestation is the nontriviality of the Welschinger invariant, which
was computed via tropical geometry. One of the most surprising manifestations of
this phenomenon is when the upper bound equals the lower bound, which is the
subject of the Shapiro Conjecture and the focus of the last five chapters.
I thank the Institut Henri Poincar´ e, where a preliminary version of these notes
was produced during a course I taught in November 2005. These notes were revised
and expanded during courses at Texas A&M University in 2007 and in 2010 and at
a lecture series at the Centre Interfacultaire Bernoulli at EPFL in 2008 and were
completed in 2011 while in residence at the Institut Mittag-Leffler with material
from a lecture at the January 2009 Joint Mathematics Meetings on the Theorem
of Mukhin, Tarasov, and Varchenko and from lectures at the GAeL conference in
Leiden in June 2009. I also thank Prof. Dr. Peter Gritzmann of the Technische
Universit¨ at M¨ unchen, whose hospitality enabled the completion of the first version
of these notes. During this period, this research was supported by NSF grants
DMS-1001615, DMS-0701059, and CAREER grant DMS-0538734. The point of
view in these notes was developed through the encouragement and inspiration of
Bernd Sturmfels, Askold Khovanskii, Maurice Rojas, and Marie-Fran¸ coise Roy and
through my interactions with the many people whose work is mentioned here, in-
cluding my collaborators from whom I have learned a great deal.
04.25.11, Djursholm, Sweden