Some Elimination Problems for Matrices

problem: Given: A field K and n variables x1, . . . , xn and m polynomials yi = pi(x1, . . . , xn) ∈ K[x1, . . . , xn] for i = 1, . . . ,m. (1) Aim: Find a presentation for the subring K[y] := K[y1, . . . , ym] of K[x] := K[x1, . . . , xn]. Invariants: The difference of m and the transcendence degree of K(y) := K(y1, . . . , ym) over K will be called the deficiency d = d(y) of the tuple y in K(x). Assumption: K perfect, so that the deficiency d(y) can be computed from the rank of the Jacobian matrix J := ( ∂yi ∂xj ) ∈ K(x) m×n, viz. d(y) = m− rank(J).