Let F be a nonformally real field, n, r be positive integers. Suppose that for any prime number p ≤ n, the quotient group F */F *p is finite. We prove that if N is large enough, then any system of r forms of degree in N variables over F has a nonzero solution. Also we show that if, in addition, F is infinite, then any diagonal form with nonzero coefficients of degree n in |F */F*n| variables is universal, i.e,, its set of nonzero values coincides with F * Bibliography: 4 titles.
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Published in Zapiski Nauchnykh Seminarov POMI, Vol. 394, 2011, pp. 209-217.
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Glazman, A.L., Zatitski, P.B., Sivatski, A.S. et al. Forms of higher degree over certain fields. J Math Sci 188, 591–595 (2013). https://doi.org/10.1007/s10958-013-1150-y
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DOI: https://doi.org/10.1007/s10958-013-1150-y