Pierce-Birkhoff conjecture is true for splines

We prove the Pierce–Birkhoff conjecture for splines, i.e., continuous piecewise polynomials of degree $d$ in $n$ variables on a hyperplane partition of ${\mathbb{R}}^n$, can be written as a finite lattice combination of polynomials. We will provide a purely existential proof, followed by a more in-depth analysis that yields effective bounds.