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Higher product Pythagoras numbers of skew fields
Velušček, Dejan
We introduce the ▫$n$▫-th product Pythagoras number ▫${\rm p}_n(D)$▫, the skew field analogue of the ▫$n$▫-th Pythagoras number of a field. For a valued skew field ▫$(D, v)$▫ where v has the property ...of preserving sums of permuted products of ▫$n$▫-th powers when passing to the residue skew field ▫${\rm k_v}$▫ and where Newton's lemma holds for polynomials of the form ▫$X^n - a$, $a \in 1 + {\rm I_v}$▫, ▫${\rm p}_n(D)$▫ is bounded above by either ▫${\rm p}_n({\rm k_v})$▫ or ▫${\rm p}_n({\rm k_v}) + 1$▫. Spherical completeness of a valued skew field ▫$(D, v)$▫ implies that the Newton's lemma holds for ▫$X^n - a$▫, ▫$a \in 1 + {\rm I_v}$▫ but the lemma does not hold for arbitrary polynomials. Using the above results we deduce that ▫${\rm p}_n (D((G))) = {\rm p}_n(D)$▫ for skew fields of generalized Laurent series.
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