Pierce–Birkhoff conjecture

In abstract algebra, the Pierce–Birkhoff conjecture asserts that any piecewise-polynomial function can be expressed as a maximum of finite minima of finite collections of polynomials. It was first stated, albeit in non-rigorous and vague wording, in the 1956 paper of Garrett Birkhoff and in which they first introduced f-rings. The modern, rigorous statement of the conjecture was formulated by and John R. Isbell, who worked on the problem in the early 1960s in connection with their work on f-rings. Their formulation is as follows:

For every real piecewise-polynomial function ${\displaystyle f\colon \mathbb {R} ^{n}\rightarrow \mathbb {R} }$, there exists a finite set of polynomials ${\displaystyle g_{ij}\in \mathbb {R} [x_{1},\ldots ,x_{n}]}$ such that ${\displaystyle f=\sup _{i}\inf _{j}(g_{ij})}$.^[1]

Isbell is likely the source of the name Pierce–Birkhoff conjecture, and popularized the problem in the 1980s by discussing it with several mathematicians interested in real algebraic geometry.^[1]

The conjecture was proved true for n = 1 and 2 by .^[2]

Local Pierce–Birkhoff conjecture[]

In 1989, provided an equivalent statement that is in terms of the of ${\displaystyle A=R[x_{1},\ldots ,x_{n}]}$ and the novel concepts of local polynomial representatives and separating ideals.

Denoting the real spectrum of A by ${\displaystyle \mathrm {Sper} A}$, the separating ideal of α and β in ${\displaystyle \mathrm {Sper} A}$ is the ideal of A generated by all polynomials ${\displaystyle g\in A}$ that change sign on ${\displaystyle \alpha }$ and ${\displaystyle \beta }$, i.e., ${\displaystyle g(\alpha )\geq 0}$ and ${\displaystyle g(\beta )\leq 0}$. Any finite covering ${\displaystyle \mathbb {R} ^{n}=\cup _{i}P_{i}}$ of closed, semi-algebraic sets induces a corresponding covering ${\displaystyle \mathrm {Sper} A=\cup _{i}{\tilde {P}}_{i}}$, so, in particular, when f is piecewise polynomial, there is a polynomial ${\displaystyle f_{i}}$ for every ${\displaystyle \alpha \in \mathrm {Sper} A}$ such that ${\displaystyle f|_{P_{i}}=f_{i}|_{P_{i}}}$ and ${\displaystyle \alpha \in {\tilde {P}}_{i}}$. This ${\displaystyle f_{i}}$ is termed the local polynomial representative of f at ${\displaystyle \alpha }$.

Madden's so-called local Pierce–Birkhoff conjecture at ${\displaystyle \alpha }$ and ${\displaystyle \beta }$, which is equivalent to the Pierce–Birkhoff conjecture, is as follows:

Let ${\displaystyle \alpha }$, ${\displaystyle \beta }$ be in ${\displaystyle \mathrm {Sper} A}$ and f be piecewise-polynomial. It is conjectured that for every local representative of f at ${\displaystyle \alpha }$, ${\displaystyle f_{\alpha }}$, and local representative of f at ${\displaystyle \beta }$, ${\displaystyle f_{\beta }}$, ${\displaystyle f_{\alpha }-f_{\beta }}$ is in the separating ideal of ${\displaystyle \alpha }$ and ${\displaystyle \beta }$.^[1]

References[]

Further reading[]

• Birkhoff, Garrett; Pierce, Richard S. (1956). "Lattice-ordered rings". Anais da Academia Brasileira de Ciências. 28: 41–69. MR 0080099. Zbl 0070.26602.
• Mahé, Louis (1984). "On the Pierce–Birkhoff conjecture". Rocky Mountain Journal of Mathematics. 14 (4): 983–986. doi:10.1216/RMJ-1984-14-4-983. MR 0773148.
• Mahé, Louis (2007). "On the Pierce–Birkhoff conjecture in three variables". Journal of Pure and Applied Algebra. 211 (2): 459–470. doi:10.1016/j.jpaa.2007.01.012. MR 2340463. Zbl 1130.13014.