Equations and complexity for the Dubois-Efroymson dimension theorem

Let R be a real closed field, let X be an irreducible real algebraic subset of R^n and let Z be an algebraic subset of X of codimension ≥ 2. Dubois and Efroymson proved the existence of an irreducible algebraic subset of X of codimension 1 containing Z. We improve this dimension theorem as follows. Indicate by μ the minimum integer such that the ideal of polynomials in R[x_1, . . . ,x_n] vanishing on Z can be generated by polynomials of degree ≤ μ. We prove the following two results: (1) There exists a polynomial P ∈ R[x_1, . . . ,x_n] of degree ≤ μ+1 such that X∩P^{−1}(0) is an irreducible algebraic subset of X of codimension 1 containing Z. (2) Let F be a polynomial in R[x_1, . . . ,x_n] of degree d vanishing on Z. Suppose there exists a nonsingular point x of X such that F(x) = 0 and the differential at x of the restriction of F to X is nonzero. Then there exists a polynomial G ∈ R[x_1, . . . ,x_n] of degree ≤ max{d, μ + 1} such that, for each t ∈ (−1, 1) \ {0}, the set {x ∈ X | F(x) + tG(x) = 0} is an irreducible algebraic subset of X of codimension 1 containing Z. Result (1) and a slightly different version of result (2) are valid over any algebraically closed field also.