Geodesic
Forms of higher degree over certain fields
Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 22, Tome 394 (2011), pp. 209-217
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Let $F$ be a nonformally real field, $n,r$ positive integers. Suppose that for any prime number $p\le n$ the quotient group $F^*/{F^*}^p$ is finite. We prove that if $N$ is big enough, then any system of $r$ forms of degree $n$ 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 $F^*$. 
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A. L. Glazman; P. B. Zatitski; A. S. Sivatski; D. M. Stolyarov. Forms of higher degree over certain fields. Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 22, Tome 394 (2011), pp. 209-217. http://geodesic.mathdoc.fr/item/ZNSL_2011_394_a6/

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