Piecewise polynomial functions
Piecewise polynomial functions
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分段多项式函数
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发表时间：
1993
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通讯作者：
Niels Schwartz
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作者：
Niels Schwartz
一个连续函数f：ℝn→ℝ是分段多项式的，如果存在一个有限半代数覆盖ℝ=M1u...Umr和多项式f1，...，fr∈ℝ[X1，...，Xn]使得f|Mi=fi|Mi对任意实闭域上的仿射半代数空间(cf.[6]。但是，如果使用与环A(Cf)相关的实闭包R(A)(也称为抽象半代数函数环)，则有一个更广泛的推广。[3]、[5]、[21])。R(A)的定义取决于A的实谱Sper(A)。[2]、[4])。实谱是从环到谱空间的函子([12])。对于环A，Sper(A)的点可以被认为是剩余域Qf(A/p)上的全序的素理想PCA。如果α表示Sper(A)的一点，则素数理想称为α的支集，表示为Supp(α)，QF(A/Supp(α))相对于α指定的总阶实闭包表示为p(α)。根据定义，R(A)是所有域p(α)的直积的子环([21，第一章)。虽然这种构造是为任意交换环定义的，但它只有在实谱不为空的情况下才有意义。
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.