For a (commutative) ring A, thepythagoras number, P(A), ofA is the smallest number n^oo such that any sum of squares in A can be expressed äs a sum of at most n squares in A. For instance, P(f?) = l, P(Fq) = 2 (if 2\q), and, by Lagrange's Theorem, P (Z) = P (G) = 4. The number P (A) is an interesting, but very delicate, arithmetic invariant of the ring A; the explicit computation of P (A) is, in general, a difficult task. Given a ring, it is often far from easy even just to decide if P (A) is finite of infinite. For some results on P(A) in the recent literature, see, for example, [P], [P2], [CEP], [HJ], [R], [PeJ, [Pe2], [EL], [Br], [Pr], etc. In some of these papers, the invariant P (A) has appeared under an assortment of other names: for instance, "Pfister dimension" in [R], "Quadratstufe" in [Pe2], and "reduced height" in [HJ] and [L]. In this paper, following [Br], [Pr], we shall call P(A) the pythagoras number and hope that, in the future, other mathematicians will adopt the same terminology.

The Pythagoras number of some affine algebras and local algebras

Introduction Main results Preliminaries Morita Equivalence Devissage Varieties of representations Generalizing Pfister's theorem Characters Detecting rationality and integrality Representation varieties: Two examples Number theory invariants All division algebras occur Appendix I. Primitive element theorems Appendix II. Hermitian categories Bibliography Index.

Invariants of boundary link cobordism

The paper deals with the first systematic study of the spaces of regular and ratinal maps between arbitrary algebraic varieties over a real closed field R. We find conditions under which these spaces are reduced to the space of Zariski locally constant maps, we investigate the finiteness properties of the subspaces of dominating regular and rational maps and, when R is the field of real numbers, we study the topology of the space of regular maps. Our results show that the mentioned map spaces are "usually" very small. The realization of such a general study has been possible thanks to the introduction of two new classes of real algebraic invariants: the curve genera and the toric genera.

Elementary structure of morphism space between real algebraic varieties

Let k be a real closed or Henselian field. A kvariety X (affine) is said to be solid if X is determined by its k points. It is shown that a k-variety is solid if and only if it contains a nonsingular k point. Another condition for solidity is given and a dimension theorem indicated. 0. Introduction. In [4] a solid k-variety is defined to be an affine kvariety which is determined by its k points. For real closed and Henselian fields with absolute value, we gave a natural condition which is necessary and sufficient for a k-variety to be solid. This condition is here extended to any Henselian field. Moreover, we also demonstrate that for any real closed or Henselian field k, a necessary and sufficient condition for a k-variety to be solid is that the variety contain a nonsingular k point. This condition is obviously insufficient when dealing with other fields such as the rationals. For example consider the curve x3+y3= 1 defined over the rationals. It has only finitely many rational points all of which are nonsingular. 1. The projection condition. Let A be a local integral domain with maximal ideal m. Let A, m be Henselian and let k be the quotient field of A. Then we wish to show that if f(x) in k[x] has a root in k, then so do certain nearby polynomials. LEMMA 1.1 (like [2, Lemma 5. 10]). If A, m is Henselian, f(x) E A [x], x e A so that (I) f'(cL)=1u; (2) for some 6 E m,rf(oa)=O mod Y 26; then f(x) has a root oc'=-ac mod ,um. PROOF [2]. Let z be a new variable and try to solve f(o+1uz)=0 for z. First expand ft(a+ z)=f(@)+4uzf'I()+4u2w(z) where w(z) e A[z] and is of degree >2. Alsof(oa)-= L2b' for some Y' E A. We want to solve Received by the editors September 14, 1971. AMS 1970 subject classifications. Primary 14G25; Secondary 12D15, 13J15.

/pdf/henselian-fields-and-solid-k-varieties-ii-2vdc8bwb1z.pdf

Henselian fields and solid $k$-varieties. II

The Nakai-Nishimura-Dubois-Efroymson dimension theorem asserts the following: "let R be an algebraically closed field or a real closed field, let X be an irreducible algebraic subset of Rn and let Y be an algebraic subset of X of codimention s>=2 (not necessarily irreducible). Then, there is an irreducible algebraic subset W of X of codimention 1 containing Y". In this paper, making use of an elementary construction, we improve this result giving explicit polynomial equations for W. Moreover, denoting by R the algebraic closure of R and embedding canonically W into projective space Pn(R), we obtain explicit upper bounds for the degree and the geometric genus of the Zariski closure of W in Pn(R). In future papers, we will use these bounds in the study of morphism space between algebraic varieties over real closed fields.

/pdf/explicit-equations-and-bounds-for-the-nakai-nishimura-dubois-1lmd3a0can.pdf

Explicit equations and bounds for the Nakai-Nishimura-Dubois-Efroymson dimension theorem

A dimension theorem for real primes

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G. Efroymson

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A dimension theorem for real primes