In most cases, the building blocks of certificates of positivity are sums of squares of polynomials. In this chapter we discuss the theory of positive polynomials and sums of squares, which underlies everything else in the book. This theory has its origins in the work of Hilbert from the late 19th century; Hilbert’s interest in these topics appears to have started when he was an officially appointed “opponent” for Minkowski’s thesis defense on July 30, 1885.

In this chapter we will follow Hilbert and work with forms (homogeneous polynomials) rather than polynomials. Recall that a ternary quartic is a form in \(\mathbb R[X,Y,Z]\) of degree 4.

A sum of squares (sos) representation for \(f \in \mathbb R[\underline{X}]\) is a global certificate of positivity, by which we mean an immediate proof that f is positive semidefinite (psd). In this chapter, we discuss finding and counting sos representations for a given f.

In 1991, as a corollary to a theorem about the multi-dimensional moment problem, Schmüdgen [1] proved that if a basic closed semialgebraic set \(K_S\) is compact, then every polynomial that is strictly positive on \(K_S\) is in the preorder \(P_S\). In other words, certificates of positivity always exist for polynomials that are strictly positive on compact \(K_S\). This result is commonly called Schmüdgen’s Positivstellensatz, and it seems to be the first example of a representation theorem without denominators for a very general class of semialgebraic sets. What is remarkable and surprising about this theorem is that it implies the existence of certificates without denominators regardless of the polynomials S that are used to define the semialgebraic set.

Many homogeneous polynomials that arise in the study of sums of squares and Hilbert's 17th problem are those formed by monomial substitutions into the arithmetic-geometric inequality. In 1989, Reznick [14] gave a necessary and sufficient condition for such a form to have a representation as a sum of squares of forms, involving the arrangement of lattice points in the simplex whose vertices were the n-tuples of the exponents used in the substitution. Further, a claim was made, and not proven, that sufficiently large dilations of any such simplex will also satisfy this condition. The aim of this short note is to prove the claim, and provide further context for the result, both in the study of Hilbert's 17th Problem and the study of lattice point simplices.

We have seen applications of certificates of positivity to problems such as optimization of polynomials on semialgebraic sets. Many of these applications use semidefinite programming or similar techniques, which means that the algorithms yield numerical output and thus may not produce an exact polynomial identity. Furthermore, in some cases an exact certificate of positivity may involve irrational numbers, which means that a numerical algorithm will never produce an exact answer. With this in mind, a natural question to ask is the following: Suppose \(f \in \mathbb Q[\underline{X}]\) such that f is sos in \(\mathbb R[\underline{X}]\), then can f be written as a sum of squares in \(\mathbb Q[\underline{X}]\)? As we shall see, in the general case, the answer is “no”. More generally, we can ask these questions replacing \(\mathbb Q\) by any subfield of \(\mathbb R\).

In this chapter, we look at polynomials that have some special structure, and discuss techniques for exploiting this structure. We discuss polynomials that are invariant under a permutation of their variables, and some whose associated Newton polytope is of a special type. For polynomials with special structure, we discuss conditions which imply or characterize when the polynomial is psd, and when it is sos.

In this chapter we discuss certificates of positivity for polytopes, which are compact basic closed semialgebraic sets in \(\mathbb R^n\) defined by linear inequalities, see Sect. 1.6. Polytopes are a particularly nice situation, since certificates of positivity exist in almost all cases, and there are several different types of representations. Any polytope can be written as a basic closed semialgebraic set \(K_S\), where \(S = \{ g_1, \ldots , g_s \}\) with each \(g_i \in \mathbb R[\underline{X}]\) linear such that no three of the \(g_i\)’s intersect in a point, see Sect. 1.6. We will always assume S is of this type.

The theorems of Schmüdgen and Putinar discussed in Chap. 7 give very satisfying answers to the question of existence of certificates of positivity for polynomials positive on a compact semialgebraic set. In this chapter, we look at certificates of positivity on noncompact semialgebraic sets. There the situation is not as well understood and there are many negative results. On the other hand, there are some surprising positive results, including examples of noncompact basic closed semialgebraic sets for which the strongest positivity property \((\ge )_M\) holds.

One of the fundamental properties of the real numbers is the Archimedean Property, an axiom introduced by Archimedes in his work on geometry. In modern language, it says that every real number is bounded above by a natural number: For \(x \in \mathbb R\) there is \(k \in \mathbb N\) so that \(x \le k\). It follows that there are no infinitesimally large or small elements in \(\mathbb R\); every element is bounded. This idea of bounded elements extends to real commutative rings in a natural way, and it turns out that boundedness is the key property that allows for the existence of denominator-free certificates of positivity.

In this chapter we look at certificates of positivity for polynomials that are positive on basic closed semialgebraic sets in \(\mathbb R\). In contrast to the higher dimensional situation, in most cases certificates of positivity exist and there are many examples of saturated preorders. We begin with the problem of finding certificates of positivity for closed intervals in \(\mathbb R\), which is well understood; here, certificates of positivity always exist. However, the existence of certificates depends on choosing the right set of generators; in contrast to Schmüdgen’s Positivstellensatz, results here are not true for any possible set of generators. Finally, we take a brief look at positivity on curves in the plane.

In this paper we study rankings induced by power indices of players in simple game models of bicameral legislatures. For a bicameral legislature where bills are passed with a simple majority vote in each house we give a condition involving the size of each chamber which guarantees that a member of the smaller house has more power than a member of the larger house, regardless of the power index used. The only case for which this does not apply is when the smaller house has an odd number of players, the larger house has an even number of players, and the larger house is less than twice the size of the smaller house. We explore what can happen in this exceptional case. These results generalize to multi-cameral legislatures. Using a standard model of the US legislative system as a simple game, we use our results to study power index rankings of the four types of players—the president, the vice president, senators, and representatives. We prove that a senator is always ranked above a representative and ranked the same as or above the vice president. We also show that the president is always ranked above the other players. We show that for most power index rankings, including the Banzhaf and Shapley–Shubik power indices, the vice president is ranked above a representative, however, there exist power indices ranking a representative above the vice president.

Using a standard model of the US legislative system as a monotonic simple game, we look at rankings of the four types of players – the president, the vice president, senators, and representatives – induced by power indices. We show that regardless of the power index used, the president is always ranked above the other players, and a senator is always ranked above the vice president and a representative. For most power index rankings, including the Banzhaf and Shapley-Shubik power indices, the vice president is ranked above a representative, however, there exist power indices ranking a representative above the vice president. Our results apply to more general yes-no voting systems.

In this paper we study rankings induced by power indices of players in simple game models of bicameral legislatures, the US legislative system, and similar legislative systems. We show that a member of the smaller house of a bicameral legislature has more power than a member of the larger house, in almost all cases. Using a standard model of the US legislative system as a simple game, we look at rankings of the four types of players - the president, the vice president, senators, and representatives. We show that regardless of the power index used, the president is always ranked above the other players, and a senator is always ranked above a representative and ranked the same as or above the vice president. For most power index rankings, including the Banzhaf and Shapley-Shubik power indices, the vice president is ranked above a representative, however, there exist power indices ranking a representative above the vice president.

In 2008, Marshall (2010) [4] settled a long-standing open problem by showing that if f(x,y)∈R[x,y] is a polynomial that is non-negative on the strip [0,1]×R, then there exist sums of squares σ(x,y),τ(x,y)∈∑R[x,y]2 such that f(x,y)=σ(x,y)+τ(x,y)(x−x2). In this paper, we generalize Marshall’s result to various strips and half-strips in the plane. Our results give many new examples of non-compact semialgebraic sets in R2 for which one can characterize all polynomials which are non-negative on the set. For example, we show that if U is a compact subset of the real line and {g1,…,gk} a specific set of generators for U as a semialgebraic set, then whenever f(x,y) is non-negative on U×R, there are sums of squares s0,…,sk such that f=s0+s1g1+⋯+skgk.

Let R[X] be the real polynomial ring in n variables. Pólya’s Theorem says that if a homogeneous polynomial p∈R[X] is positive on the standard n-simplex Δn, then for sufficiently large N all the coefficients of (X1+⋯+Xn)Np are positive. We give a complete characterization of forms, possibly with zeros on Δn, for which there exists N so that all coefficients of (X1+⋯+Xn)Np have only nonnegative coefficients, along with a bound on the N needed.

If a real polynomial f in n variables can be written as a sum of squares of real polynomials, then clearly f must take only nonnegative values in R. This simple, but powerful, fact and generalizations of it underlie a large body of theoretical and computational results concerning positive polynomials and sums of squares. An explicit expression of f as a sum of squares is a certificate of positivity for f , i.e., a polynomial identity which gives an immediate proof of the positivity of of f on R. In recent years, much work has been devoted to the study of certificates of positivity for polynomials. In this paper we will give an overview of some recent results in the theory and practice of positivity and sums of squares, with detailed references to the literature. By “theory”, we mean theoretical results concerning the existence of certificates of positivity. By “practice”, we mean work on computational and algorithmic issues, such as finding certificates of positivity for a given polynomial. For the most part, we restrict results to those in a real polynomial ring. This is somewhat misleading, since it is impossible to prove most of the results for polynomials without using a more abstract approach. For example, in order to obtain a solution to Hilbert’s 17th problem, it was necessary for Artin (along with Schreier) to first develop the theory of ordered fields! The reader should keep in mind that underneath the theorems in this paper lie the elegant and beautiful subjects of Real Algebra and Real Algebraic Geometry, among others. The subject of positivity and sums of squares has been well-served by its expositors. There are a number of books and survey articles devoted to various aspects of the subject. Here we mention a few of these that the interested reader could consult for more details and background on the topics covered in this paper, as well as related topics that are not ∗Department of Mathematics and Computer Science, Emory University, Atlanta, GA 30322. Email: vicki@mathcs.emory.edu.

Schm\"udgen's Theorem says that if a basic closed semialgebraic set K = {g_1 \geq 0, ..., g_s \geq 0} in R^n is compact, then any polynomial f which is strictly positive on K is in the preordering generated by the g_i's. Putinar's Theorem says that under a condition stronger than compactness, any f which is strictly positive on K is in the quadratic module generated by the g_i's. In this note we show that if the g_i's and the f have rational coefficients, then there is a representation of f in the preordering with sums of squares of polynomials over Q. We show that the same is true for Putinar's Theorem as long as we include among the generators a polynomial N - \sum X_i^2, N a natural number.

Pólya's Theorem says that if p is a homogeneous polynomial in n variables which is positive on the standard n-simplex, and F is the sum of the variables, then for a sufficiently large exponent N, FN * p has positive coefficients. Pólya's Theorem has had many applications in both pure and applied mathematics; for example it provides a certificate for the positivity of p on the simplex. The authors have previously given an explicit bound on N, determined by the data of p; namely, the degree, the size of the coefficients and the minimum value of p on the simplex. In this paper, we extend this quantitative Pólya's Theorem to non-negative polynomials which are allowed to have simple zeros at the corners of the simplex.