An Ideal Separating Extension of Affine Space

In affine space the set of solutions to a system of polynomial equations does not uniquely determine the system. We extend affine space so that the solutions (in the extension) to a system of equations uniquely determines the system. 1. Statement of the problem In particular, for field R let elements of R[x] = R[x1, ..., xn] act on the set of power series R[[T ]] = R[[T1, ..., Tn]] by way of the linear extension of the action x(T) = Tm−k for m − k in N and x(T) = 0 for m − k / ∈ N. For the ideal I ∈ (the set of all ideals in R[x]) let N(I) = {f(T ) ∈ R[[T ]] | P (x)f(T ) = 0∀P (x) ∈ I} (which we call the “generalized solutions” of I). We also set Z(I) = {r ∈ R | P (r) = 0∀P (x) ∈ I}. The injection ρ : R → R[[T ]] given by ρ(r) = ρ(r1, ..., rn) = ∑ rT k is such that ρ(Z(I)) ⊂ N(I) and so ρ maps the affine solutions to a system of equations into the generalized solutions of that system. Let N be the set of all N(I) for I ∈ . We show that N is a one-to-one order reversing bijection from to N which implies that the generalized solutions to a system of equations uniquely determines the system. 2. Introduction Let denote the set of all ideals in the polynomial ring R[x] ≡ R[x1, x2, ..., xn] where R is a field, let Z(I) ≡ {r = (r1, ..., rn) ∈ R|P (r) = 0 for all P (x) ∈ I} be the algebraic set corresponding to any I ∈ , and let ζ ≡ {Z(I)|I ∈ } be the set of all the algebraic sets in R. We will say that R “separates” a set of ideals ̂ ⊂ if we have I1 = I2 whenever we have Z(I1) = Z(I2) for I1, I2 ∈ ̂. While working on problems in invariant theory, Hilbert proved the powerful theorem (called Nullstellensatz) that rad( ) is separated in R when R is an algebraically closed field (where rad(I) ≡ {g = g(x) ∈ R[x]|∃m ∈ N so that g ∈ I} is the radical of the ideal I and where rad( ) is the set of all radical ideals). When n = 1 the radical ideals are of the form 〈P (x)〉 where P (x) has distinct roots. Hilbert’s Nullstellensatz implies that 〈P (x)〉 has as many solutions as the degree of P (x). Consequently, Hilbert’s Nullstellensatz theorem has been called an n-dimensional generalization of the fundamental theorem of algebra. Received by the editors April 24, 2003 and, in revised form, March 9, 2005. 2000 Mathematics Subject Classification. Primary 14xx, 13xx. c ©2007 American Mathematical Society Reverts to public domain 28 years from publication 3071 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 3072 PAUL S. PEDERSEN A consequence of Hilbert’s Nullstellensatz is that when R is algebraically closed there is a one to one order reversing correpondence between the lattice of radical ideals and the lattice of algebraic sets of R (both partially ordered by inclusion). In this paper we describe an extension of affine space having enough points to separate all ideals. In fact we show that to separate all ideals it is sufficient to consider an extension of affine space which only separates the zero ideals (those having Z(I) = {0}). An added feature of the method is that the field need not be algebraically closed. To do this we use the R linear space of power series R[[T ]] = R[[T1, T2, ..., Tn]] on which we allow the elements of R[x] to operate by the linear extension of the action (for k, j ∈ N) given by x(T ) = T k−j if k − j ∈ N and by x(T ) = 0 if k − j / ∈ N. We use (1) N(I) = {f(T ) ∈ R[[T ]]|P (x)f(T ) = 0∀P (x) ∈ I} to denote the algebraic nullspace associated with the ideal I and we let N denote the set of all algebraic nullspaces of R[[T ]]. We embed R in R[[T ]] by way of the injection (2) ρ(r) = ρ(r1, ..., rn) = ∑