Piecewise polynomial functions

A continuous function f:ℝn→ℝ is piecewise polynomial if there are a finite semi-algebraic cover ℝ = M1U...UMr and polynomials f1,...,fr ∈ ℝ[X1,...,Xn] such that f ∣Mi = fi∣Mi for every i=1,...,r. Virtually the same definition can be used to explain the notion of a piecewise polynomial function on an affine semi-algebraic space over an arbitrary real closed field (cf. [6]. But there is a much more sweeping generalization if one uses the real closure R(A) (also called the ring of abstract semi-algebraic functions) associated with a ring A (cf. [3], [5], [21]). The definition of R(A) depends on the real spectrum Sper(A) of A (cf. [2], [4]). The real spectrum is a functor from rings to spectral spaces ([12]). For a ring A the points of Sper(A) may be thought of as prime ideals pCA together with a total order on the residue field qf(A/p). If α denotes a point of Sper(A) then the prime ideal is called the support of α, denoted by supp(α), and the real closure of qf(A/supp(α)) with respect to the total order specified by α is denoted by p(α). By definition, R(A) is a subring of the direct product of all the fields p(α) ([21, Chapter I). Although this construction is defined for arbitrary commutative rings it is of significance only if the real spectrum is not empty.