Sums of squares in nonreal commutative rings. (English) Zbl 1401.11080
Since 50 years there appeared a lot of papers on sums of squares in a (commutative) ring \(R\) and on the invariants \(s(R)\) (level) and \(p(R)\) (Pythagoras number). We should perhaps mention the important papers by M. Peters [Math. Ann. 195, 309–314 (1972; Zbl 0217.04201); J. Reine Angew. Math. 268/269, 318–323 (1974; Zbl 0287.10034)], R. Baeza [Math. Ann. 207, 121–131 (1974; Zbl 0248.13031); Math. Ann. 215, 13–21 (1975; Zbl 0287.13006); Quadratic forms over semilocal rings. Springer, Cham (1978; Zbl 0382.10014); Arch. Math. 33, 226–231 (1979; Zbl 0421.10015)], Z. D. Dai et al. [Bull. Am. Math. Soc., New Ser. 3, 845–848 (1980; Zbl 0435.10016)] and D. W. Hoffmann [J. Am. Math. Soc. 12, No. 3, 839–848 (1999; Zbl 0921.11018)] in the list of references. In the present paper the authors introduce two new invariants of \(R\) which they call “metalevel” and “hyperlevel”. They allow a more systematic treatment and slightly sharper inequalities between the (four) invariants of \(R\). Several of the earlier proofs can now be simplified. Nevertheless the paper also contains a lot of new examples where the “exact values” of the invariants could not be determined.
Reviewer: Albrecht Pfister (Mainz)
MSC:
| 11E25 | Sums of squares and representations by other particular quadratic forms |
| 11E04 | Quadratic forms over general fields |
| 13F20 | Polynomial rings and ideals; rings of integer-valued polynomials |
| 13F25 | Formal power series rings |
Citations:
Zbl 0217.04201; Zbl 0287.10034; Zbl 0248.13031; Zbl 0287.13006; Zbl 0382.10014; Zbl 0421.10015; Zbl 0435.10016; Zbl 0921.11018References:
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