Charles N. Delzell

Case Distinctions are Necessary for Representing polynomials as Sums of Squares

Publisher Summary This chapter provides an overview of the history of logical aspects of Hilberts 17th problem. In his book on the foundations of geometry, Hilbert described those problems in plane geometrical construction that can be solved by means of only his five groups of axioms, can always be carried out by the use of straightedge and gauge. He gave two algebraic characterizations of the set of points so constructible, in terms of their Cartesian coordinates (f 1 (x), f 2 (x)), where the given points are expressed as rational functions of the parameters x = (x 0 , …, x n ) ɛ ℝ n+1 . The second of his two characterizations was a necessary and sufficient condition—namely that f i (x) be a totally real-algebraic number for all x ɛ ℚ n+1 . The chapter discusses two logical points concerning the finiteness theorem: (1) intuitionistic considerations and (2) the terminology finiteness.

Real Algebraic Geometry and Ordered Structures

We show how one may sometimes perform singular ambient surgery on the complex locus of a real algebraic curve and obtain what we call a floppy curve. A floppy curve is a certain kind of singular surface in CP (2), more general than the complex locus of a real nodal curve. We derive analogs for floppy curves of known restrictions on real nodal curves. In particular we derive analogs of Fielder’s congruence for certain nonsingular curves and Viro’s inequalities for nodal curves which generalize those of Arnold and Petrovskii for nonsingular curves. We also obtain a determinant condition for certain curves which are extremal with respect to some of these equalities. One may prohibit certain schemes for real algebraic curves by prohibiting the floppy curves which result from singular ambient surgery. In this way, we give a new proof of Shustin’s prohibition of the scheme 1 < 2 ∐ 1 < 18 >> for a real algebraic curve of degree eight.

A continuous and rational solution to Hilbert’s 17th problem and several cases of the Positivstellensatz

From the Positivstellensatz we construct a continuous and rational solution for Hilbert’s 17th problem and for several cases of the Positivstellensatz. The solutions are obtained using an especially simple method.

Positive Polynomials on Semialgebraic Sets

In this chapter we study improvements of Theorem 5.2.9, which, for polynomials strictly positive on a bounded semialgebraic set of the form W ℝ(h 1 ,..., h s ), gives a canonical representation involving products of the h i ’s. It will be shown that in general, not all products are needed. In many cases the “linear” representation (0.5) of the Introduction can actually be achieved.

Continuous Sums of Squares of Forms

We give a continuous representation of positive semidefinite (psd) n -ary quadratic forms over an ordered field as sums of (almost n ! e ) nonnegatively-weighted squares of linear forms. This answers a question of Kreisel, who noticed in 1980 that (already for n=2) the usual “completion-of-square” process gives a discontinuous representation. For n =2 J.F. Adams has recently reduced the required number of continuous summands to 2, but only over Euclidean ordered fields. We also show that any universal representation of psd quadratic forms as sums of squares of quadratic forms must be discontinuous at ( X 2 + Y 2 ) 2 .

Continuous Pythagoras numbers for rational quadratic forms

Abstract We introduce the concept of the continuous Pythagoras number P c ( S ) of a subset S of a commutative topological ring to be, roughly, the least number m ≤ ∞ such that the set of sums of squares of elements of S can be represented as sums of m squares of elements of S , by means of m continuous functions. Heilbronn had already shown that P c ( Q ) = 4. Letting L n ( F ) be the set of linear n -ary forms over the field F , we show that P c (L n ( R )) = n . We then allow continuously varying nonnegative rational “weights” on the m square summands. If these continuous weight functions and the continuous functions giving the coefficients of the m linear forms, are required to be Q -rational functions of the coefficients of the given positive semidefinite quadratic forms, then we show that P c (L 1 ( R )) = 1 and P c (L n ( R )) = ∞ for n > 1. However, if only the product of the weight functions and the coefficient functions is required to be continuous, then n ≤ P c ( L n ( R )) n ! e ] (where e is the base of the natural logarithms) and 2 P c ( L 2 ( R )); we conjecture that n P c ( L n ( R )) also for n > 2. On the other hand, if these weight functions and coefficient functions are required only to be rational in the weaker sense of taking rational values at rational arguments, then P c ( L 2 ( Q )) = 2, and we conjecture that P c ( L n ( Q )) = n also for n > 2.

Properties of Model Classes

In this chapter we wish to study properties of model classes. By a model class we mean the class of all models of an axiom system Σ.

Model Theory of Several Algebraic Theories

In this chapter we investigate a series of interesting algebraic theories for the properties of completeness, model completeness and quantifier elimination. Not only do we treat the standard examples that have already been frequently treated in the extant literature, but we shall place special value on the theory of valued fields. Since valuation theory does not belong in the standard repertoire of an algebra course, we shall first discuss the necessary concepts and theorems in detail, in Section 4.3. Thereafter we develop special cases (Sections 4.4 and 4.5), and finally the model theory of Henselian valued fields. The goal of this presentation is, among other things, a treatment of a purely number theoretic problem – Artin’s conjecture – in Theorem 4.6.5.

First-Order Logic

In this chapter we introduce a calculus of logical deduction, called first-order logic, that makes it possible to formalize mathematical proofs. The main theorem about this calculus that we shall prove is Godel’s completeness theorem (1.5.2), which asserts that the unprovability of a sentence must be due to the existence of a counterexample. From the finitary character of a formalized proof we then immediately obtain the Finiteness Theorem (1.5.6), which is fundamental for model theory, and which asserts that an axiom system possesses a model provided that every finite subsystem of it possesses a model.

Sums of 2mth Powers

In this chapter we show that many of the results obtained so far still remain true when we replace “sums of squares” by “sums of 2mth powers.” Clearly, such sums inherit all the properties of “positivity,” but in a more refined way. Thus “generalizing” a result from sums of squares to sums of 2mth powers actually represents a strengthening.