Received: 6 November 2016 / Revised: 23 March 2017 / Accepted: 27 March 2017
The only answer to Question 1.1 in published literature (obtained via a MathOverflow enquiry [auniket2010]) seems to be a class of counterexamples constructed by [Bayer2002] for $ n \geq 32$ using Nagata's counterexample to Hilbert's fourteenth problem from [Nagata1965] and Weitzenböck's theorem ([Weitzenböck1932]) on finite generation of invariant rings. After an earlier version of this article appeared on arXiv, however, Wilberd van der Kallen communicated to me a simple counterexample for $ n = 3$:
\begin{align} y \mapsto y + x^3,\quad z \mapsto z + x^2 \label{neenaction} \end{align} | (1.1) |
A variant of Example 1.2 in fact gives a counterexample to Question 1.1 for $ n = 2$:
Since Question 1.1 holds for $ n = 1$ (see e.g. assertion (1) of Theorem 1.5), Example 1.3 gives a complete answer to Question 1.1. In this article we consider a natural variant of Question 1.1: denote the subrings of $ \cc[x_1, \ldots, x_n]$ in Question 1.1 by $ R_1, R_2$, and their intersection by $ R$.
Note that in each of Examples 1.2 and 1.3 the ring $ R_1$ is not integrally closed, so that they do not apply to Question 1.4. Our findings are compiled in the following theorem.
Assertions (1) and (2) follow in a straightforward manner from results of [Zariski1954] and [Schröer2000]. Assertion (3) is the main result of this article: the subrings $ R_1$ and $ R_2$ from our examples are easy to construct, and our proof that they are finitely generated is elementary; however the proof of non-finite generation of $ R_1 \cap R_2$ uses the theory of key forms (introduced in [Mondal2016a]) of valuations centered at infinity on $ \cc^2$.
Finite generation of subalgebras of polynomial algebras has been well studied, see e.g. [Gale1957], [Nagata1966], [Evyatar and Zaks1970], [Eakin1972], [Nagata1977], [Wajnryb1982], [Gilmer and Heinzer1985], [Amartya2008] and references therein. One of the classical motivations for these studies has been Hibert's fourteenth problem. Indeed, as we have mentioned earlier, Bayer's counterexamples to Question 1.1 for $ n \geq 32$ were based on Nagata's counterexamples to Hilbert's fourteenth problem. Similarly, the construction of Example 1.2 is a special case of a result of [Bhatwadekar and Daigle2009] on the ring of invariants of the additive group $ (\cc, +)$. Our interest in Questions 1.1 and 1.4 however comes from two other aspects: compactifications of $ \cc^n$ and the moment problem on semialgebraic subsets of $ \rr^n$—this is explained in Sect. 2.
In Sect. 2 we explain our motivations to study Question 1.1. In Sect. 3 we prove assertions (1) and (2) of Theorem 1.5, and in Sect. 4 we prove assertion (3). Theorem 4.1 gives the general construction of our counterexamples to Question 1.4 for $ n = 3$, and Example 4.2 contains a simple example. Appendix A gives an informal introduction to key forms used in the proof of Theorem 4.1, and Appendix B contains the proof of a technical result used in the proof of Theorem 4.1.
Our original motivation to study Question 1.1 comes from construction of projective compactifications of $ \cc^n$ via degree-like functions . More precisely, given an affine variety $ X$ over a field $ \kk$, a degree-like function on the ring $ \kk[X]$ of regular functions on $ X$ is a map $ \delta: \kk[X] \to \zz \cup \{-\infty\}$ which satisfies the following properties satisfied by the degree of polynomials:
The graded ring associated with $ \delta$ is
\begin{align} \kxdelta &:= \dsum_{d \geq 0} \{f \in \kk[X]: \delta(f) \leq d\} \nonumber\\ &\cong \sum_{d \geq 0} \{f \in \kk[X]: \delta(f)\leq d\}t^d \subseteq \kk[X][t] \label{kxdelta} \end{align} | (2.1) |
It is straightforward to check that the maximum of finitely many degree-like functions is also a degree-like function, and taking the maximum is one of the basic ways to construct new degree-like functions (see e.g. [Mondal2014], Theorem 4.1). For example, an $ n$-dimensional convex polytope $ \scrP \subset \rr^n$ with integral vertices and containing the origin in its interior determines a degree-like function on $ \kk[x_1, x_1^{-1}, \ldots, x_n, x_n^{-1}]$ defined as follows: \begin{eqnarray*} \delta_\scrP\left(\sum a_\alpha x^\alpha\right) := \inf\{d\in \zz: d \geq 0,\ \alpha \in d\scrP\ \text{for all}\ \alpha \in \zz^n\ \text{such that}\ a_\alpha \neq 0\} \end{eqnarray*} It is straightforward to see that $ \delta_\scrP$ satisfies properties (iv) and (v), so that it determines a projective completion $ X_\scrP$ of the torus $ \nktorus$. It turns out that $ X_\scrP$ is precisely the toric variety corresponding to $ \scrP$. Moreover, $ \delta_\scrP$ is the maximum of some other 'simpler' degree-like functions determined by facets of $ \scrP$—see Fig. 1 for an example.
The preceding discussion suggests that the following is a fundamental question in the theory of degree-like functions:
In the scenario of Question 2.1, identifying $ \kxdeltaone$ and $ \kxdeltatwo$ with subrings of $ \kk[X][t]$ as in (2.1) implies that $ \kxdelta = \kxdeltaone \cap \kxdeltatwo$. Consequently, in the case that $ \kk = \cc$ and $ X$ is the affine space $ \cc^n$, Question 2.1 is a special case of Question 1.1, and our counterexamples to Question 1.4 are in fact counterexamples to this special case with $ X = \cc^2$.
Given a closed subset $ S$ of $ \rr^n$, the $ S$-moment problem asks for characterization of linear functionals $ L$ on $ \rr[x_1, \ldots, x_n]$ such that $ L(f)= \int_S f\, \mathrm{d} \mu$ for some (positive Borel) measure $ \mu$ on $ S$. Classically the moment problem was considered on the real line ($ n = 1$): given a linear functional $ L$ on $ \rr[x]$, a necessary and sufficient condition for $ L$ to be induced by a positive Borel measure on $ S \subseteq \rr$ was shown to be
In particular, the classical examples show that $ \emptyset$, $ \{x\}$, $ \{x, 1 - x\}$ solves the moment problem respectively for $ \rr$, $ [0, \infty)$, $ [0, 1]$. In the case that $ S$ is a basic semialgebraic set, i.e. $ S$ is defined by finitely many polynomial inequalities $ f_ 1 \geq 0, \ldots, f_s \geq 0$, [Schmüdgen1991] proved that $ \{f_1, \ldots, f_s\}$ solves the $ S$-moment problem provided $ S$ is compact. On the other hand, if $ S$ is non-compact, then it may happen that no finite set of polynomials solves the moment problem for $ S$ (see e.g. [Kuhlmann and Marshall2002], [Powers and Scheiderer2001]). Netzer associated (see e.g. [Mondal and Netzer2014], Section 1) a natural filtration $ \{\scrB_d(S): d \geq 0\}$ on the polynomial ring determined by $ S$: \begin{align*} \scrB_d(S) &:= \{f \in \rr[x_1, \ldots, x_n]: f^2 \leq g\ \text{on}\ S\ \text{for some}\ g \in \rr[x_1, \ldots, x_n],\\ &\quad\ \deg(g) \leq 2d\} \end{align*} In other words, $ \scrB_d(S)$ is the set of all polynomials which 'grow on $ S$ as if they were of degree at most $ d$'. The graded algebra corresponding to the filtration is \begin{align*} \scrB(S) := \dsum_{d \geq 0} \scrB_d(S) \cong \sum_{d \geq 0} \scrB_d(S)t^d \subseteq \rr[x_1, \ldots, x_n, t] \end{align*} where $ t$ is a new indeterminate.
It is straightforward to produce open semialgebraic sets $ S$ which satisfies the assumption of Theorem 2.3. E.g. a standard tentacle is a set \begin{eqnarray*} \left\{ (\lambda^{\omega_1}b_1,\ldots,\lambda^{\omega_n}b_n)\mid \lambda \in \rr, \lambda \geq 1, b\in B\right\} \end{eqnarray*} where $ \omega := (\omega_1, \ldots, \omega_n) \in \zz^n$ and $ B\subseteq (\rr{\setminus}\{0\})^n$ is a compact semialgebraic set with nonempty interior; we call $ \omega$ the weight vector corresponding to the tentacle. If $ S$ is a finite union of standard tentacles with weights $ \omega_1, \ldots, \omega_k \in \zz^n$, then it is not too hard to see that
Let $ p, q_1, \ldots, q_k, \omega_1, \omega_2$ be as in conditions (A)–(D) of Sect. 4.1. Pick nonzero $ a_1, \ldots, a_k \in \rr$ and define $ f_+(x), f_-(x)$ as in (4.1) and (4.2) . Note that as opposed to Sect. 4.1, here $ f_+(x)$ and $ f_-(x)$ are polynomials over real numbers . For each $ i \in \{+,-\}$, pick positive real numbers $ c_{i,1} < c_{i,2}$ and define \begin{align*} S_i := \{(x,y) \in \rr^2: x \geq 1,\ c_{i,1} \leq x^{\omega_2/\omega_1} ( y - f_i(x)) \leq c_{i,2}\} \end{align*} Let $ \delta_+, \delta_-, R_+, R_-$ be as in Sect. 4.1. For $ i \in \{+,-\}$, [Mondal and Netzer2014] ([Mondal and Netzer2014], Lemma 4.3) implies that $ f(x,y) \in \scrB_d(S_i)$ iff $ \delta_i(f) \leq p\omega_1d$. It follows that the map $ \phi: t \mapsto t^{p\omega_1}$ maps $ \scrB(S_i) \into R_i$. It is straightforward to check that $ R_+, R_ -, R_+ \cap R_-$ are integral over $ \scrB(S_+), \scrB(S_-), \scrB(S_+) \cap \scrB(S_-)$ respectively. Lemma 3.2 and Theorem 4.1 then imply that $ \scrB(S_+)$ and $ \scrB(S_-)$ are finitely generated over $ \rr$, but $ \scrB(S_+ \cup S_-) = \scrB(S_-) \cap \scrB(S_+)$ is not, even though $ \scrB_0(S_+ \cup S_-) = \rr$. Figure 2 depicts a pair of $ S_+$ and $ S_-$ corresponding to Example 4.2.
In this section we prove assertions (1) and (2) of Theorem 1.5. The proof remains valid if $ \cc$ is replaced by an arbitrary algebraically closed field. Moreover, if $ k$ is a field with algebraic closure $ \bar k$, then a subring $ R$ of $ k[x_1, \ldots, x_n]$ is finitely generated over $ k$ iff $ R \otimes_k \bar k$ is finitely generated over $ \bar k$; this, together with the preceding sentence, implies that assertions (1) and (2) of Theorem 1.5 remain true if $ \cc$ is replaced by an arbitrary field. We use the following results in this section.
Recall the notation from Theorem 1.5. In this section we write $ L$ for the field of fractions of $ R$ and $ \bar R_j$ for the integral closure of $ R_j$ in its field of fractions, $ j = 1, 2$. Moreover, we write $ R'_j := \bar R_j \cap L$, $ j = 1,2$.
Assume w.l.o.g. $ \trdeg_\cc(L) = 1$. Theorem 3.3 implies that $ R'_1$ is finitely generated as a $ \cc$-algebra. Let $ C$ be the unique non-singular projective curve over $ \cc$ such that the field of rational functions on $ C$ is $ L$. Then $ C'_1 := \spec R'_1$ is isomorphic to $ C {\setminus} \{x_1, \ldots, x_k\}$ for finitely many points $ x_1, \ldots, x_k \in C$. Then the local rings $ \sheaf_{C,x_j}$ of $ C$ at $ x_j$'s are the only one dimensional valuation rings of $ L$ not containing $ R'_1$. Let $ \bar R$ be the integral closure of $ R$ in $ L$. Since $ \bar R \subseteq R'_1$, Lemma 3.1 implies that \begin{eqnarray*} \bar R = R'_1 \cap \sheaf_{C,x_{j_1}} \cap \cdots \cap \sheaf_{C,x_{j_s}} \end{eqnarray*} for some $ j_1, \ldots, j_s \in \{1, \ldots, k\}$. Then $ \bar R$ is the ring of regular functions on $ C {\setminus} \{x_j: j\not\in \{j_1, \ldots, j_s\}\}$, and is therefore finitely generated over $ \cc$. Lemma 3.2 then implies that $ R$ is finitely generated over $ \cc$.
Let $ L$ be the field of fraction of $ R$. Due to assertion (1) we may assume $ \trdeg_\cc(L) = 2$. Theorem 3.3 implies that $ R'_1$ and $ R'_2$ are finitely generated over $ \cc$. Let $ X_i := \spec R'_i$ and $ \bar X_i$ be a projective compactification of $ X_i$, $ i = 1,2$. Let $ \bar X$ be the closure in $ \bar X_1 \times \bar X_2$ of the graph of the birational correspondence $ X_1 \dashrightarrow X_2$ induced by the identification of their fields of rational functions, and $ \tilde X$ be the normalization of $ \bar X$. For each $ i$, let $ \pi_i: \tilde X \to \bar X_i$ be the natural projection and set $ U_i := \pi_i^{-1}(X_i)$.
The assumption that $ R_i$'s are integrally closed together with Claim 3.5 and Theorem 3.4 imply that $ R = R'_1 \cap R'_2 = \Gamma(U_1 \cup U_2, \sheaf_{\tilde X})$ is finitely generated over $ \cc$, as required. $ \square$
In this section we prove assertion (3) of Theorem 1.5. In Sect. 4.1 we describe the construction of counterexamples to Question 1.4 for $ n =3$, and in Sects. 4.2 and 4.3 we prove that these satisfy the required properties.
Let $ p, q_1, \ldots, q_k$ be integers such that
and let $ \omega_1,\omega_2$ be relatively prime positive integers such that
\begin{equation} f_+(x) :=x^p + \sum_{j=1}^k a_j x^{-q_j}\label{fspssps}\\ \end{equation} | (4.1) |
\begin{equation} f_-(x) := f_+(-x) = - x^p + \sum_{j=1}^k (-1)^{q_j} a_j x^{-q_j} \label{fspssps1} \end{equation} | (4.2) |
\begin{align} R_i := \cxydeltai = \sum_{d \geq 0} \{g \in \cc[x,y]: \delta_i(g) \leq d\}t^d \subseteq \cc[x,y,t] \end{align} | (4.3) |
Assertion (3) of Theorem 1.5 follows from Theorem 4.1 below.
Let $ \Delta_d := \{f \in \cc[x,y]: \delta_i(f) \leq d, \ i = 1, 2\} $, so that $ R_+ \cap R_- = \sum_{d \geq 0} \Delta_d t^d$. Then
We prove assertion (1) of Theorem 4.1 only for $ R_+$, since the statement for $ R_-$ follows upon replacing each $ a_j$ to $ (-1)^{q_j}a_j$.
The fact that $ R_+$ is integrally closed follows from the observation that $ \delta_+(g^k) = k\delta_+(g)$ for each $ g \in \cc[x,y]$ and $ k \geq 0$, i.e. $ \delta_+$ is a subdegree in the terminology of [Mondal2010] (see e.g. [Mondal2010], Proposition 2.2.7). We give a proof here for the sake of completeness.
Lemma 4.4 shows that $ \Rplus$ is integrally closed. Now we show that $ \Rplus$ is finitely generated over $ \cc$. Set $ q_0 := 0$ and define
\begin{align} \gplus{0} &:= y - x^p\\ \gplus{j} &:= x^{q_j}(y - x^p - \sum_{i=1}^j a_{i} x^{-q_i}),\quad 1 \leq j \leq k,\\ \omegapluss{j} &:= \deltaplus(\gplus{j}) = \begin{cases} -\omega_1(q_{j+1} - q_j) &\text{if}\ 0 \leq j \leq k-1, \\ -\omega_2 &\text{if}\ j = k. \end{cases} \end{align} | (4.4) |
\begin{align} \omegaplus(F) \geq \deltaplus(\piplus(F)) \label{omegadeltaplus} \end{align} | (4.5) |
\begin{align} S/\tilde J_+\cong \cc[x,x^{-(q_1-q_0)}, \ldots, x^{-(q_k - q_{k-1})}, z_k] \label{SspsJspssps} \end{align} | (4.6) |
Since $ \Gamma$ is a finitely generated subsemigroup of $ \zz^{k+3}$, Corollary 4.7 proves assertion (1) of Theorem 4.1.
Let $ u,v, \xi$ be indeterminates. Let
\begin{align} \phi(u,\xi) := f_+(u^{1/2}) + \xi u^{-\omega_2/(2\omega_1)} = u^{p/2} + \sum_{j=1}^k a_ju^{-q_j/2} + \xi u^{-\omega_2/(2\omega_1)} \end{align} | (4.7) |
Now note that \begin{align*} v^2|_{v = \phi(u,\xi)} &= u^p + 2a_1u^{(p-q_1)/2} + \cdots + 2a_ku^{(p-q_k)/2} + 2\xi u^{p/2 - \omega_2/(2\omega_1)} + \ldt \end{align*} where $ \ldt$ denotes terms with degree in $ u$ smaller than \begin{eqnarray*} \epsilon := p/2 - \omega_2/(2\omega_1) \end{eqnarray*} Note that $ \epsilon \geq 0$ due to defining property (D) of $ \omega_1, \omega_2$. Define
\begin{align} h_j &= \begin{cases} v^2 - u^p &\text{if}\ j = 0,\\ h_{j-1} - 2a_ju^{-q_j/2}v &\text{if}\ 1 \leq j \leq k\ \text{and } q_j \text{ is even,} \\ h_{j-1} - 2a_ju^{(p-q_j)/2} &\text{if}\ 1 \leq j \leq k\ \text{and } q_j \text{ is odd.} \end{cases} \end{align} | (4.8) |
It then follows that $ u, v, h_0, \ldots, h_k$ is the sequence of key forms of $ \eta$—see Appendix A for an informal discussion of key forms, and ([Mondal2016a], definition 3.16) for the precise definition. Property (C) of $ q_1, \ldots, q_k$ implies that $ h_k$ is not a polynomial. This, together with observation (a) and [Mondal and Netzer2014] ([Mondal and Netzer2014], theorem 4.13 and proposition 4.14) implies (see Appendix A.4) that
Since $ R = \cxydelta$ is integral over $ \cc[u,v]^\eta$, observations (c)–(f) imply assertions (2)–(5) of Theorem 4.1. $ \square$
The simplest of the degree-like functions on $ \cc[x,y]$ are weighted degrees : given a pair of relatively prime integers $ (\omega_1, \omega_2) \in \zz^2$, the corresponding weighted degree $ \omega$ is defined as follows: \begin{align*} \omega\left(\sum_{\alpha, \beta}c_{\alpha,\beta}x^{\alpha}y^\beta\right) &:= \max\{\alpha\omega_1 + \beta\omega_2: c_{\alpha,\beta} \neq 0\} \end{align*}
Assume $ \omega_1$ and $ \omega_2$ are positive. Then the weighted degree $ \omega$ can also be described as follows: take the one dimensional family of curves $ C_\xi := \{(x,y): y^{\omega_1} - \xi x^{\omega_2} = 0\}$ parametrized by $ \xi \in \cc$. Each of these curves has one place at infinity , i.e. its closure in $ \pp^2$ intersects the line at infinity on $ \pp^2$ at a single point, and the germ of the curve is analytically irreducible at that point. Then for each $ f \in \cc(x,y)$, $ \omega(f)$ is simply the pole of $ f|_{C_\xi}$ at the unique point at infinity on $ C_\xi$ for generic $ \xi \in \cc$.
Now consider the family of curves $ D_\xi := \{(x,y): y^2 - x^3 - \xi x^2 = 0\}$, again parametrized by $ \xi \in \cc$. Each $ D_\xi$ also has one place at infinity, and therefore defines a degree-like function $ \eta$ on $ \cc(x,y)$ defined as in the preceding paragraph: $ \eta(f)$, where $ f$ is a polynomial, is the pole of $ f|_{D_\xi}$ at the unique point at infinity on $ D_\xi$ for generic $ \xi \in \cc$. Then it is not hard to see that
\begin{align} f = \sum_{\alpha_0, \alpha_1, \alpha_2} c_\alpha x^{\alpha_0}y^{\alpha_1}(y^2 - x^3)^{\alpha_2} \end{align} | (4.9) |
Both $ \omega$ from Appendix A.1 and $ \eta$ from Appendix A.2 are divisorial semidegrees on $ \cc[x,y]$ - these are degree-like functions $ \delta$ on $ \cc[x,y]$ such that there is an algebraic compactification $ \bar X$ of $ \cc^2$ and an irreducible curve $ E \subseteq \bar X {\setminus} \cc^2$ such that for each $ f \in \cc[x,y]$, $ \delta(f)$ is the pole of $ f$ along $ E$. For a divisorial semidegree $ \delta$, starting with $ g_0 := x, g_1 := y$, one can successively form a finite sequence of elements $ g_0, \ldots, g_{l+1} \in \cc[x,x^{-1},y]$, $ l \geq 0$, such that
A lot of information of a divisorial semidegree $ \delta$ can be recovered from its key forms. The results that we use in the proof of Theorem 4.1 follow from [Mondal and Netzer2014] ([Mondal and Netzer2014], theorem 4.13 and proposition 4.14), and are as follows: if $ g_{l+1}$ is the last key form of $ \delta$, then
\begin{align}\label{gensgfspscondition} \delta(f) = \max\{\delta_1(f), \ldots, \delta_k(f)\} \end{align} | (4.10) |
Let $ A \subseteq B$ be $ \kk$-algebras which are also integral domains. Assume $ B$ is integral over $ A$ and the quotient field $ L$ of $ B$ is a finite separable extension of the quotient field $ K$ of $ A$.
\begin{align} P(T) := T^d + \sum_e g_e T^{d-e}\label{Pspstsps} \end{align} | (4.11) |
\begin{align} \delta_i(g_e) = \eta'_i(g_e) \leq e\eta'_i(f). \label{deltaspsispsgspsesps} \end{align} | (4.12) |
\begin{align} \delta_i(g_e) \leq ed'\quad \text{for all}\ i,\; 1 \leq i \leq m. \label{deltaspsispsgspsespsspsagain} \end{align} | (4.13) |