SUM OF SQUARES DECOMPOSITION FOR SYMMETRIC POLYNOMIAL INEQUALITIES A thesis presented to the faculty of San Francisco State University '- ifi In partial fulfillment of > The Requirements for The Degree MATH * £ 6 ^ Master of Arts In Mathematics by Logan Coe San Francisco, California August 2017 Copyright by Logan Coe 2017 CERTIFICATION OF APPROVAL I certify that I have read SUM OF SQUARES DECOMPOSITION FOR SYMMETRIC POLYNOMIAL INEQUALITIES by Logan Coe and that in my opinion this work meets the criteria for approving a thesis sub­ mitted in partial fulfillment of the requirements for the degree: Master of Arts in Mathematics at San Francisco State University. Serkan Hosten Professor of Mathematics Pierre Langlois Professor of Mathematics Lipika Deka Associate Professor of Mathematics SUM OF SQUARES DECOMPOSITION FOR SYMMETRIC POLYNOMIAL INEQUALITIES Logan Coe San Francisco State University 2017 We develop a new approach to verifying inequalities between symmetric functions. By reorganizing desired inequalities between these symmetric functions we use non­ negativity to verify the inequality. Our approach uses symmetry adapted semidefi- nite programming to show that a given symmetric function is not only nonnegative, but in fact a sum of squares. By designing an invariant semidefinite program we re­ duce our problem size using the natural symmetries in SOS semidefinite programs. Using these methods we validate known inequalities corresponding to elementary and power sum symmetric functions. We also expand on the conjecture that Schur and complete homogeneous symmetric functions have similar inequalities. We test our approach on all four symmetric functions in cases where hand computation would be impossible. Based on our results we conjecture that a Sum of Squares de­ composition exists for every inequality including the two conjectured inequalities. I certify that the Abstract is a correct representation of the content of this thesis. ACKNOWLEDGMENTS The author thanks Professor Federico Ardila for helping inspire this re­ search as well as Gina Karunaratne for her contribution to it in the early stages of this work. The author also thanks San Francisco State University for the funding needed to run optimization problems through Matlab. Lastly the author thanks his advisor Serkan Hosten for all his patience, explanations, and inspirations. TABLE OF CONTENTS 1 Introduction............................................................................................................. 1 2 Background.......................................................................................................... 7 2.1 Symmetric Functions.................................................................................... 7 2.2 Sum of S q u a re s..................................................................................................18 2.3 Semidefinite Programming M ethods.............................................................. 26 3 SOS D ecom positions.................................................................................................33 4 Reduction ....................................................................................................................42 4.1 Representation T h e o ry .....................................................................................42 4.2 Symmetry Adaptation .....................................................................................57 5 Conclusion.................................................................................................................... 81 6 A p p e n d ix ....................................................................................................................83 Bibliography .....................................................................................................................84 LIST OF TABLES Table vii LIST OF FIGURES Figure 2.1 Graph of a Convex Function 1 Chapter 1 Introduction Applications of symmetric functions are found in many branches of mathematics such as enumerative combinatorics, group theory, Lie algebras, algebraic geome­ try, and optimization. Inequalities between symmetric functions is a classic topic. Perhaps the most famous example is the inequality between the Arithmetic and Geometric means which is shown and proved in [2], In order to fully understand the goal of this thesis, we must first have a funda­ mental understanding of symmetric functions. A function of multiple variables is called symmetric if its value remains the same under all permutations of the variables , i.e., a function f(xu ...,xn) is symmetric if f(x^ i),. .., xa(n)) = f(xu for all a € Sn where Sn is the symmetric group on n elements. For example the function f(xi, x2, x3) = XiX2 + X1.X3 + x2x3 is symmetric because addition and multiplication are commutative. Or in other 2 words [xix2 + 0:1X3 + x2x3) = (x2xi + x2x3 + X1X3) = (rr2a;3 + x2xi + x3xi) and so on. A symmetric function is homogeneous if each term has the same degree, which is true for the example above. The symmetric functions that we work with are defined by partitions of nonneg­ ative integers. A partition of a nonnegative integer n is a decreasing sequence of nonnegative integers less than or equal to n where the sum of the terms is equal to n. For instance the set of all partitions of 3 is {(3,0,...), (2,1,0,...), (1,1,1,0,...)} which for simplicity will be denoted {3; 2, 1; 1, 1, 1}. A partial ordering on partitions of the same integer can be used to define a partial order on special homogeneous symmetric functions of the same degree. The focus of our research is to show that methods in Optimization can be used to verify this relationship. The set of homogeneous symmetric functions of degree r in n variables forms a vector space and has several commonly used bases. In particular, we are interested in the elementary symmetric functions, power sum symmetric functions, (complete) homogeneous symmetric functions, and Schur functions defined in Section 2.1. These are defined by partitions of r. For instance, if the number of variables n — 3 and the degree of the polynomial r — 3, then the elementary symmetric function 3 corresponding to the partition 1,1,1 of 3 is ei,i,i(xi, x2, x3) = (xi + x 2 + x3)3 For each basis listed above, the term-normalized symmetric function G\{x) as­ sociated to the element g\(x) is defined to be G\{x)r r„\ —- 9x{x) 9\(!>•••> 1)’ where A is a partition of r. So, the term-normalized symmetric function for the elementary symmetric function eititi(xi, x2, x3) is t-t , x ew (x1,x2,x3) fxi + x2 + x3\ 3 E«(*•■*».*»> = ^(IXTT = I 5 J • Inequalities between two term-normalized symmetric functions F(x) and G(x) have been studied in [3], where for a fixed n F(x) < G(x) or G(x) — F(x) > 0 if F(a) < G(a) for all a € R>0. 4 T heorem 1.1 ([3]). E\(x) > E^x), x > 0 <= > A ■< n P\(x) < Pn(x), x > 0 <*=>■ A ■< /j, Hx(x) < H,(x), x > 0 A ■< n S\(x) < Sv(x), x > 0 =► A d: H where x G R>0, E, P, H , and S are the term normalized Elementary, Power Sum, Homogeneous, and Schur functions and A and fi are partitions of the same nonnega­ tive integer r. The symbol ^ refers to a partial order which we will define in Section 2. 1. The similarities between these theorems led to Conjecture ?? in [3] that the converses of the last two partial orders are also true. Here, we take a new approach to verifying the inequalities using semidefinite pro­ gramming. A semidefinite program is a specific form of optimization problem that we can solve using [8],[7]. One way to show that a polynomial with real coefficients is nonnegative is to show that it is a sum of squares (SOS) of other polynomials. This is just a sufficient condition as there are some nonnegative polynomials that are not SOS (for instance the Motzkin Poynomial x\x\ + x\x\ — 3x\x\ + 1). Using methods shown in [1] we discover whether a given polynomial can be decomposed as an SOS by solving a semidefinite optimization problem. The problem lies in finding 5 a positive semidefinite matrix that can be used to build our polynomial. Finding such a matrix is achievable through solving a semidefinite program. The positive definiteness of this matrix allows us to construct the polynomial as a sum of squares. We use other work found in [5], to take advantage of the symmetries within our optimization problem. This involves using representation theory to simplify our semidefinite programs for faster computation. Even in small cases we see massive reductions of problem size. For instance in Section 4.2, we see the ultimate reduction of a 210 variable problem to just 11 variables. Using all of these methods we verify the results from Theorem 1.1 for n = 2 and 3 and r = 2 and 3. This shows that our approach to verifying theorem 1.1 and testing the conjecture is viable. W ith this evidence, we formed the following conjecture of our own: Conjecture 1 .1 . Let F{xi,... ,xn) and G(xi,... ,xn) be symmetric functions from the families used in Theorem 1.1. Then F{xi,... ,x„) > G(xi,... ,xn), for x > 0 for the listed cases if and only if F(y\,..., y%) — G(yf,..., y„) is an SOS. In Chapter 2 we define the tools needed to understand the problem we are trying to solve, as well as the tools we use to solve it. Here we discuss topics such as sym­ metric functions, integer partitions, partial ordering, and optimization. In Chapter 3, using concepts from [1] to approach the problem of determining whether a given multivariate polynomial is a sum of squares using semidefinite programming, we show how particular inequalities stated