Real Piecewise Algebraic Variety

Abstract

We give definitions of real piecewise algebraic variety and its dimension. By using the techniques of real radical ideal, P-radical ideal, affine Hilbert polynomial, Bernstein-net form of polynomials on simplex, and decomposition of semi-algebraic set, etc., we deal with the dimension of the real piecewise algebraic variety and real Nullstellensatz in $C^\mu$ spline ring.