Given a polynomial equation, how many roots does it have? The fundamental theorem of algebra implies that if we count the “right” way — including multiplicities and complex roots — then the number of roots is equal to the degree of the polynomial. And Bezout’s Theorem lets us count how many roots there are in systems of polynomial equations, if we count in the same way. But as beautiful as these theorems are, I always feel like they are a bit of a lie, because if someone hands me a polynomial and asks how many roots there are they probably don’t want the answer counted with multiplicity — and they probably are mostly interested in the solutions over the real numbers. If you are like me then at some point you learned Descrates Rule Of Signs, which implies (among other things) that a polynomial in a single variable with m nonzero terms has at most 2m–1 real roots, but you never thought much deeper about the question than that.
But Frank Sottile has. These are the types of questions that Sottile considers in his new book Real Solutions to Equations From Geometry: given a system of multivariate polynomial equations, how many solutions are there over the real numbers? This book is less concerned with algorithms for how to find the roots than in how to determine how many roots there are in a given range. If your interests lie more towards the algorithmic then there are other books on that topic. In particular, I have had the pleasure of reviewing two such books — Algorithms in Real Algebraic Geometry by Basu, Pollack, and Roy and Solving Polynomial Equations edited by Dickenstein and Emiris — in this space in previous years.
The opening chapter of Sottile’s book gives a wonderfully written and largely elementary introduction to the subject, discussing various bounds and results that are known and in most cases elaborated on in the later chapters. One example of such a bound comes from the “fewnomial” bound proven by Khovanskii in 1980, which says that a system of n polynomials in n variables that has a total of n+l+1 distinct monomials has at most 2a(n+1)l+n positive solutions, where a is the binomial coefficient l+nC2. However, this bound is far from sharp as shown by a 2003 theorem due to Li, Rojas, and Wang which proves that two trinomials in two variables have at most 5 positive solutions rather than the 248,832 that would be allowed by Khovanskii’s bound. Sottile himself has given further improvements to Khovanskii’s bound, and some of these are the subjects of later chapters of the book.
While giving explicit answers for the number of real solutions to general systems is quite difficult, much more can be said if one restricts their interest to systems that arise in natural ways from geometry. One example of such a result shows that if one is given four lines in three-dimensional space that are sufficiently general then there will exist two additional lines that meet all four of these lines. This is the first example of the Shapiro Conjecture, which Sottile formulates in terms of the solutions to inverse Wronskian maps — a topic which is too technical to go into in this book review but which Sottile readily explains in a very clear way. Five of the book’s fourteen chapters are devoted to the Shapiro Conjecture — what it says in certain special cases, how it can be proven in special cases, and how would could generalize it beyond Grassmannians. One example of such a formulation states that a rational function with only real critical points is equivalent (up to isomorphism) to a rational function with only real zeroes — a theorem which can be proven in elementary means, a proof due to Eremenko and Gabrielov that Sottile presents in his book.
While the geometric hypotheses that allow Sottile and others to give more precise results are in some ways very special, they are also not all that unreasonable, as they show up in several types of applications that are discussed in the book, including polynomials arising from the theory of posets and from the Schubert Calculus as well as questions in enumerative algebraic geometry. As an example, Sottile shows that given eight general points in the real projective plane there are at least eight real rational cubics that go through them and for some choices of the eight points there are as many as twelve.
Real Solutions to Equations from Geometry is a very well-written book that discusses some very exciting and modern algebraic geometry that has roots in questions that can be easily formulated even at the level of high school students. While the book gets quite technical at times, Sottile manages to include many examples and pictures and keep the exposition clear and light. The level of exposition is especially high for a book about an area of mathematics that is active and ongoing, and Sottile manages to highlight a number of open (or partially open) questions throughout. I learned quite a bit from the book and I would recommend it to those looking to learn more about the subject.
Darren Glass is an Associate Professor of Mathematics at Gettysburg College whose primary mathematical interests include number theory and Galois theory. He can be reached at email@example.com.