Polynomial origami (Q1895633)

This paper is mainly concerned with describing the subalgebras of the polynomial algebra \(k[x]\) over a field \(k\). The first point is that every epic into \(k[x]\) is surjective. This follows (indirectly) from the following unpublished theorem from \textit{A. Ogus}: If \(K\) is an algebraically closed field, then \(K[x]\) has no maximal proper subalgebras except the two obvious types, (i) \(\{f \in K[x] : f(p) = f(q)\}\) and (ii) \(\{f \in K[x] : f'(p) = 0\}\). -- Apart of Ogus' proof which makes heavy use of algebraic geometry the authors give a much simpler proof. They show (further) that for algebraically closed \(K\) and subalgebras \(A \subset B \subset K[x] \), the codimension of \(A\) in \(B\) is the length of a maximal chain of subalgebras from \(B\) to \(A\); and for any field \(k\), every subalgebra of \(k[x]\) is defined in the style of (i) and (ii) above, by linear equations on the functionals \(f \mapsto f(p)\), \(f \mapsto f'(p)\), and in general \(f \mapsto f^{(n)} (p)/n!\). For all \(k\), for \(A \subset B \subset k[x]\), if \(A\) is maximal in \(B\) then \(A\) has finite codimension in \(B\). Describing the subalgebras of \(k[x]\), the authors make use of a ``size'' of a subalgebra \(A\) of \(k[x]\) which can be measured by two natural numbers \(r(A)\), \(d(A)\), as follows. The set \(\deg A\) of degrees of polynomials in \(A\) is a submonoid of \(\mathbb{N}\), and therefore it is most of an arithmetic progression \(r \mathbb{N}\), precisely \(r \mathbb{N} \backslash D\) for a finite set \(D\). The authors call \(r = r(A)\) the rarity of \(A\) and they call the size \(d(A)\) of \(D\) the defect of \(A\). \(r(A)\) and \(d(A)\) are related to how the functions \(k \mapsto k\) named by polynomials in \(A\) map \(k\). This introduces the ``origami'' of the title. An equivalence relation \(\mathbb{R} \subset k \times k\) is called a fold if it is the relation ``for all \(f \in A\), \(f(x) = f(y)\)'' for some \(A \subset k[x]\), which may be taken to be \(\mathbb{R}^\perp = \{f \in k[x] : f(p) = f(q)\) for all \((p,q)\) in \(\mathbb{R}\}\). The subsets \(\mathbb{R}^\perp\) are subalgebras, which are called fold algebras. There is a Galois connection here, \(A^\perp\) being \(\{(p,q) : f(p) = f(q)\) for all \(f \in A\}\). The authors relate \(r\) and \(d\), for a fold algebra \(A\) over an algebraically closed field of characteristic 0, to the fold \(A^\perp\): almost all \(A^\perp\)-equivalence classes have size \(r(A)\), and the sum of the deviations from that size is \(1 + r(d - 1)\). (The deviation of \(n\) from \(r\) is \(n - r\); it may be negative.) Two other results are surprising. Note that the definition of \(A^\perp\) above works for any subset \(A\) of \(k[x]\). Call the depth of a fold \(A^\perp\), and of the fold algebra \(A^{\perp \perp}\), the minimum size of a set that \(S\) such that \(S^\perp = A^\perp\). It turns out that every fold over any field has depth at most 3; and every fold over the rational field \(\mathbb{Q}\) has depth at most 1. There are other interesting results concerning cyclic closures and difference quotients.