Lecture Notes

The lecture notes provided below are preliminary and an ongoing work.

1 Introduction

Review of Convexity and Linear Programming

2 PSD Matrices

Semidefinite Programming

3 Binary Optimization

Bounds: Goemans-Williamson and Nesterov

Linearly Constrained Problems

4 Review: Groups, Rings, Fields

Polynomials and Ideals

5 Univariate Polynomials

Root Bounds and Sturm Sequences

Counting Real Roots


Sum of Squares

Positive Semidefinite Matrices

6 Resultants



The set of Nonnegative Polynomials

7 Hyperbolic Polynomials

SDP Representability

8 SDP Representability

Convex Sets in R2 \n \nHyperbolicity and the Lax Conjecture

Relating SDP-representable Sets and Hyperbolic Polynomials


9 Binomial Equations

Newton Polytopes

The Bézout and BKK Bounds

Application: Nash Equilibria

10 Nonegativity and Sums of Squares

Sums of Squares and Semidefinite Programming

Applications and Extensions

Multivariate Polynomials

Duality and Density

11 SOS Applications


Bridging the Gap

12 Recovering a Measure from its Moments

A Probabilistic Interpretation

Duality and Complementary Slackness

Multivariate Case

Density Results

13 Polynomial Ideals

Algebraic Varieties

Quotient Rings

Monomial Orderings

14 Monomial Orderings

Gröbner Bases

Applications and Examples

Zero-dimensional Ideals

15 Zero-dimensional Ideals

Hilbert Series

16 Generalizing the Hermite Matrix

Parametric Versions

SOS on Quotients

17 Infeasibility of Real Polynomial Equations


The Zero-dimensional Case


18 Quantifier Elimination


Cylindrical Algebraic Decomposition (CAD)

19 Certificates

Psatz Revisited

Copositive Matrices and Pólya’s Theorem

Positive Polynomials

20 Positive Polynomials

Schmüdgen’s Theorem

21 Groups and their Representations

Algebra Decomposition

22 Sums of Squares Programs and Polynomial Inequalities (PDF)

Course Info

Learning Resource Types

notes Lecture Notes