Elimination over skew fields
Paul M. Cohn · 2019
Over a commutative field there is a process known as the elimination of quantifiers, which is very useful. The simplest instance is provided by the resultant of two polynomials, whose vanishing is a criterion for the two polynomials to have a common zero. More precisely, given two polynomials f = ∑ a i x i , g = ∑ b j x j over a field k , there exists a polynomial R ( a , b ) in the coefficients a i , b j such that R( a , b ) = 0 if and only if f and g have a common zero in some extension field of k. Here we must be careful to allow ∞ as a possible zero; alternatively we could replace x by a projective coordinate.