Vol. 97, No. 2, 1981

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Second note on Artin’s solution of Hilbert’s 17th problem. Order spaces

D. W. Dubois

Vol. 97 (1981), No. 2, 357–371

We consider real varieties in Kn∕k, where K∕k is maximally ordered and k is dense in K. Our principal results are:

Theorem 1. Assume V is irreducible. A. The closure of the set of all simple points equals the set of all central points — z V is central if some order of k(V )∕k contains every function which is positive at z. B. If f + r is totally positive in k(V )∕k for every positive r in k, then f itself is totally positive.

Theorem 2. For a semi-algebraic set S in Kn defined by polynomial relations bj(x) = 0, gi(x) > 0 (1 i,j m), define B = (b1,,bm), G = {g1,,gm}. Then every irreducible component of V k(B) contains central points on S if and only if () for 0 < pi k, gij G, ai k[X], ipiΠjgij ai2 R√ --
B implies every ai R√ --

Mathematical Subject Classification 2000
Primary: 12D15
Secondary: 12J15, 14G30
Received: 25 June 1980
Published: 1 December 1981
D. W. Dubois