Abstract. Techniques based on sums of squares appear promising as a general approach to the universal theory of reals with addition and multiplication, i.e. verifying Boolean combinations of equations and inequalities. A particularly attractive feature is that suitable ‘sum of squares ’ certificates can be found by sophisticated numerical methods such as semidefinite programming, yet the actual verification of the resulting proof is straightforward even in a highly foundational theorem prover. We will describe our experience with an implementation in HOL Light, noting some successes as well as difficulties. We also describe a new approach to the univariate case that can handle some otherwise difficult examples. 1 Verifying nonlinear formulas over the reals Over the real numbers, there are algorithms that can in principle perform quantifier elimination from arbitrary first-order formulas built up using addition, multiplication and the usual equality and inequality predicates. A classic example of such a quantifier elimination equivalence is the criterion for a quadratic equation to have a real root: ∀a b c. (∃x. ax 2 + bx + c = 0) ⇔ a = 0 ∧ (b = 0 ⇒ c = 0) ∨ a � = 0 ∧ b 2 ≥ 4ac

Let k be a field of characteristic zero, V a smooth, positive-dimensional, quasiprojective variety over k, and Q a nonempty divisor on V. Let K be the function field of V, and A ⊂ K the semilocal ring of Q. We prove the Diophantine undecidability of: (1) A, in all cases; (2) K, when k is real and V has a real point; (3) K, when k is a subfield of a p-adic field, for some odd prime p. To achieve this, we use Denef’s method: from an elliptic curve E over Q, without complex multiplication, one constructs a quadratic twist E of E over Q(t), which has Mordell-Weil rank one. Most of the paper is devoted to proving (using a theorem of R. Noot) that one can choose f in K, vanishing at Q, such that the group E (K) deduced from the field extension Q(t) ∼ → Q(f) ֒ → K is equal to E (Q(t)). Then we mimic the arguments of Denef (for the real case) and of Kim and Roush (for the p-adic case).

Abstract. We generalize a question of Büchi: Let R be an integral domain and k ≥ 2 an integer. Is there an algorithm to solve in R any given system of polynomial equations, each of which is linear in the k−th powers of the unknowns? We examine variances of this problem for k = 2, 3 and for R a field of rational functions of characteristic zero. We obtain negative answers, provided that the analogous problem over Z has a negative answer. In particular we prove that the generalization of Büchi’s question for fields of rational functions over a real-closed field F, for k = 2, has a negative answer if the analogous question over Z has a negative answer. 1

This is a translation of a paper [5] I wrote in 1971, and may help for Parimala’s course. Evidently completely outdated, but still may be useful. I changed some notation so as to be compatible with the course.

Abstract. Let K be a totally real number field with Galois closure L. We prove that if f ∈ Q[x1,..., xn] is a sum of m squares in K[x1,..., xn], then f is a sum of 4m · 2 [L:Q]+1([L: Q] + 1 2 squares in Q[x1,..., xn]. Moreover, our argument is constructive and generalizes to the case of commutative K-algebras. This result gives a partial resolution to a question of Sturmfels on the algebraic degree of certain semidefinite programing problems. 1.

I work on a wide range of problems that arise from other areas of mathematics and the physical sciences. Currently, I am focused on using mathematical and computational tools to solve basic problems in theoretical neuroscience, and in this regard, I have begun collaborations with scientists at the Redwood Center for Theoretical Neuroscience and mathematicians at U.C. Berkeley. I am also interested in theoretical questions involving semidefinite programming, optimization, and computational algebra. The following is a description of several interrelated lines of research in which I will actively participate in the coming years. The first three sections contain very brief discussions of topics related to theoretical neuroscience that I have only begun exploring in recent months. The final sections describe more theoretical studies that I have been investigating in recent years and therefore contain more detailed descriptions. 1. Sparse coding and compressed sensing Sparse coding refers to the process of representing a real vector input (such as an image) as a sparse linear combination of an overcomplete set of vectors (called a sparse basis). Here, overcomplete refers to the fact that there are many more vectors in the sparse basis

Let q be a quadratic form over a field K of characteristic different from 2. We investigate the properties of the smallest positive integer n such that −1 is a sums of n values represented by q in several situations. We relate this invariant (which is called the q-level of K) to other invariants of K such as the level, the u-invariant and the Pythagoras number of K. The problem of determining the numbers which can be realized as a q-level for particular q or K is studied. We also observe that the q-level naturally emerges when one tries to obtain a lower bound for the index of the subgroup of non-zero values represented by a Pfister form q. We highlight necessary and/or sufficient conditions for the q-level to be finite. Throughout the paper, special emphasis is given to the case where q is a Pfister form.

Suppose that f is a real univariate polynomial of degree D with exactly 4 monomial terms. We present a deterministic algorithm of complexity polynomial in logD that, for most inputs, counts the number of real roots of f. The best previous algorithms have complexity super-linear in D. We also discuss connections to sums of squares and A-discriminants, including explicit obstructions to expressing positive definite sparse polynomials as sums of squares of few sparse polynomials. Our key theoretical tool is the introduction of efficiently computable chamber cones, which bound regions in coefficient space where the number of real roots of f can be computed easily. Much of our theory extends to n-variate(n+3)-nomials.

We prove that the recursively enumerable relations over a polynomial ring R[t], where R is the ring of integers in a totally real number field, are exactly the Diophantine relations over R[t].