An algebraic proof of the Borsuk-Ulam theorem for polynomial mappings

Abstract.Upper bounds for the number of variables necessary to imply the existence of an m-dimensional linear variety on the intersection of r cubic hypersurfaces over local and global fields are given. (dl,..., dr; m) such that every projective variety V c p n which is the common zero set of r forms in n + 1 variables of degrees dl,..., dr with coefficients in F necessarily contains a projective m-dimensional linear F space provided n >f OF (dl,..., dr; m), and the conclusion fails for smaller n. Since it is easily demonstrated that 7Q~ (d) = 7p (d) exists for the p-adic field ~p, and indeed 7p (d) ~< d 2 + 1 for all p and d (see [7]), it follows that (bp (dl .... , d,.; m) = cb~p (all,..., dr; m) exists. The argument as presented by R. Brauer did not provide estimates for q)p (dl,...,dr;m) and simple minded calculations suggested by his argument would provide upper bounds on q~p of excessively high exponential order. Over the last 40 years considerable effort has been expended on obtaining bounds for q~p when m = 0. We mention only a few of the results that have been obtained. Let q~p (dl,... , dr) = ~p(dl,..., dr; 0

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