4 problems

Let $f\colon\mathbb{R}^n\to\mathbb{R}$ be a semialgebraic spline, meaning a continuous piecewise polynomial function on a semialgebraic partition of $\mathbb{R}^n$. Pierce–Birkhoff…

Let $PWP(n)_u$ be the set of continuous piecewise-polynomial functions from $[0,1]^n$ to $[0,1]$, let $ISD(n)_u$ be the set of continuous functions on $[0,1]^n$ expressible as fini…

Let $n\geq 1$. The free, commutative, real $\ell$-algebra (equivalently, $\ell$-group) on $n$ generators is compared with the $\ell$-group of real-valued functions that are continu…

Let $A$ be a regular ring and let $\alpha,\beta\in\operatorname{Sper} A$. For a piecewise-polynomial function $f\in PW(A)$, let $f_\alpha,f_\beta\in A$ be local representatives at…