Publication: Sums of squares of linear forms: the quaternions approach
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2006
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Abstract
Let A = k[y] be the polynomial ring in one single variable y over a field k. We discuss the number of squares needed to represent sums of squares of linear forms with coefficients in the ring A. We use quaternions to obtain bounds when the Pythagoras number of A is ≤ 4. This provides bounds for the Pythagoras number of algebraic curves and algebroid surfaces over k.
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E. Becker: Hereditarily-Pythagorean fields and orderings of higher level. Monografías de Matemática [Mathematical Monographs], 29. Rio de Janeiro: Instituto de Matemática Pura e Aplicada, 1978.
J. Bochnak, M. Coste, M. F. Roy: Real Algebraic Geometry. Ergeb. Math. 36. Berlin Heidelberg New York: Springer-Verlag, 1998.
J. W. S. Cassels: On the representation of rational functions as sums of squares. Acta Arith. 9 (1964) 79–82.
M. D. Choi, Z. D. Dai, T. Y. Lam, B. Reznick: The Pythagoras number of some affine algebras and local algebras. J. reine Angew. Math. 336 (1982) 45–82.
M. D. Choi, T. Y. Lam, B. Reznick: Real zeros of positive semidefinite forms. I. Math. Z. 171, 1–26 (1980).
D. Z. Djoković: Hermitean matrices over polynomial rings. J. Algebra 43 (1976) 359–374.
J. F. Fernando: On the Pythagoras numbers of real analytic rings. J. Algebra 356 (2004) 2663–2684.
J. F. Fernando, J. M. Ruiz, C. Scheiderer: Sums of squares in real rings. Trans. AMS 243 (2001) 321–338.
J. F. Fernando, J. M. Ruiz, C. Scheiderer: Sums of squares of linear forms. To appear in Math. Research Letters.
T.W. Hungerford: Algebra. Graduate Text in Math. 73. Berlin Heidelberg New York: Springer Verlag, 1974.
J. M. Ruiz: On pythagorean real algebroid curves. Rocky Mountain J. Math. 14 (1984) 899–901.
C. Scheiderer: On sums of squares in local rings. J. reine angew. Math. 540 (2001), 205–227.