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When is the Intersection of Two Finitely Generated Subalgebras of a Polynomial Ring Also Finitely Generated?

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Abstract

We study two variants of the following question: “Given two finitely generated \(\mathbb {C}\)-subalgebras \(R_1, R_2\) of \(\mathbb {C}[x_1, \ldots , x_n]\), is their intersection also finitely generated?” We show that the smallest value of n for which there is a counterexample is 2 in the general case, and 3 in the case that \(R_1\) and \(R_2\) are integrally closed. We also explain the relation of this question to the problem of constructing algebraic compactifications of \(\mathbb {C}^n\) and to the moment problem on semialgebraic subsets of \(\mathbb {R}^n\). The counterexample for the general case is a simple modification of a construction of Neena Gupta, whereas the counterexample for the case of integrally closed subalgebras uses the theory of normal analytic compactifications of \(\mathbb {C}^2\) via key forms of valuations centered at infinity.

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Notes

  1. The integral closure of a subring R of a ring S is the set of all elements \(x \in S\) which satisfies an equation of the form \(x^d + \sum _{i=1}^d a_ix^{d-i} = 0\) for some \(d > 0\) and \(a_1, \ldots , a_d \in R\). A domain is integrally closed if it itself is its integral closure in its field of fractions.

  2. See Appendix A.1 for a discussion of weighted degrees.

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Acknowledgements

I would like to thank Pierre Milman—the mathematics of this article was worked out while I was his postdoc at University of Toronto. I would also like to thank Wilberd van der Kallen for providing Example 1.2, and the referees for some suggestions which significantly improved the quality of the exposition of this article. The first version of this article has been written up during the stay at the Weizmann Institute as an Azrieli Fellow, and the later versions at the University of the Bahamas.

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Correspondence to Pinaki Mondal.

Appendices

Appendix A: Key Forms: An Informal Introduction

1.1 Appendix A.1

The simplest of the degree-like functions on \(\mathbb {C}[x,y]\) are weighted degrees: given a pair of relatively prime integers \((\omega _1, \omega _2) \in \mathbb {Z}^2\), the corresponding weighted degree \(\omega \) is defined as follows:

$$\begin{aligned} \omega \left( \sum _{\alpha , \beta }c_{\alpha ,\beta }x^{\alpha }y^\beta \right)&:= \max \{\alpha \omega _1 + \beta \omega _2: c_{\alpha ,\beta } \ne 0\} \end{aligned}$$

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 \(\xi \in \mathbb {C}\). Each of these curves has one place at infinity, i.e. its closure in \(\mathbb {P}^2\) intersects the line at infinity on \(\mathbb {P}^2\) at a single point, and the germ of the curve is analytically irreducible at that point. Then for each \(f \in \mathbb {C}(x,y)\), \(\omega (f)\) is simply the pole of \(f|_{C_\xi }\) at the unique point at infinity on \(C_\xi \) for generic \(\xi \in \mathbb {C}\).

1.2 Appendix A.2

Now consider the family of curves \(D_\xi := \{(x,y): y^2 - x^3 - \xi x^2 = 0\}\), again parametrized by \(\xi \in \mathbb {C}\). Each \(D_\xi \) also has one place at infinity, and therefore defines a degree-like function \(\eta \) on \(\mathbb {C}(x,y)\) 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 \(\xi \in \mathbb {C}\). Then it is not hard to see that

  • \(\eta (x) = 2\), \(\eta (y) = 3\), \(\eta (y^2 - x^3) = 4\),

  • Given an expression of the form

    $$\begin{aligned} f = \sum _{\alpha _0, \alpha _1, \alpha _2} c_\alpha x^{\alpha _0}y^{\alpha _1}(y^2 - x^3)^{\alpha _2} \end{aligned}$$
    (13)

    where \(0 \le \alpha _1 < 2\) and \(\alpha _2 \ge 0\), one has

    $$\begin{aligned} \eta (f) = \max \{ 2\alpha _0 + 3\alpha _1 + 4\alpha _2: c_\alpha \ne 0\} \end{aligned}$$

1.3 Appendix A.3

Both \(\omega \) from Appendix A.1 and \(\eta \) from Appendix A.2 are divisorial semidegrees on \(\mathbb {C}[x,y]\) - these are degree-like functions \(\delta \) on \(\mathbb {C}[x,y]\) such that there is an algebraic compactification \(\bar{X}\) of \(\mathbb {C}^2\) and an irreducible curve \(E \subseteq \bar{X} {\setminus } \mathbb {C}^2\) such that for each \(f \in \mathbb {C}[x,y]\), \(\delta (f)\) 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 \(g_0, \ldots , g_{l+1} \in \mathbb {C}[x,x^{-1},y]\), \(l \ge 0\), such that

  • for each \(i = 1, \ldots , l\), \(g_{i+1}\) is a simple ‘binomial’ in \(g_0, \ldots , g_i\),

  • \(\delta (g_{i+1})\) is smaller than its ‘expected value’, and

  • every polynomial f in (xy) has an expression in terms of \(g_0, \ldots , g_{l+1}\) such that \(\delta (f)\) can be computed from that expression from only the knowledge of \(\delta (g_0), \ldots , \delta (g_{l+1})\).

The key forms of weighted degrees are simply xy, and the key forms of \(\eta \) from Appendix A.2 are \(x, y, y^2 - x^3\) (since \(\eta (x) = 2\) and \(\eta (y) = 3\), the ‘expected value’ of \(\eta (y^2 - x^3)\) should have been 6, whereas its actual value is 4).

1.4 Appendix A.4

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 Netzer (2014, theorem 4.13 and proposition 4.14), and are as follows: if \(g_{l+1}\) is the last key form of \(\delta \), then

  1. (i)

    The following are equivalent:

    1. (1)

      \(\delta (f) > 0\) for every non-constant polynomial f on \(\mathbb {C}[x,y]\),

    2. (2)

      either \(\delta (g_{l+1}) > 0\), or \(\delta (g_{l+1}) = 0\) and \(g_{l+1}\) is not a polynomial.

  2. (ii)

    The following are equivalent:

    1. (1)

      \(\mathbb {C}[x,y]^{\delta }\) is not finitely generated over \(\mathbb {C}\),

    2. (2)

      \(\delta (g_{l+1}) \ge 0\) and \(g_{l+1}\) is not a polynomial.

  3. (iii)

    Assume \(\mathbb {C}[x,y]^{\delta }\) is not finitely generated over \(\mathbb {C}\).

    1. (1)

      if \(\delta (g_{l+1}) > 0\), then \(L_d := \{f \in \mathbb {C}[x,y]: \delta (f) \le d\}\) is a finite dimensional vector space over \(\mathbb {C}\) for each \(d \ge 0\).

    2. (2)

      if \(\delta (g_{l+1}) =0\), then there exists \(d > 0\) such that \(L_d\) is infinite dimensional over \(\mathbb {C}\).

Appendix B: Integral Closure of the Graded Ring of a Degree-Like Function

Definition B.1

Let \(\mathbb {K}\) be a field and A be a \(\mathbb {K}\)-algebra. A degree-like function \(\delta \) on A is called a semidegree if \(\delta \) satisfies condition (ii) of degree-like functions (see Sect. 2.1) always with an equality. We say that \(\delta \) is a subdegree if there are finitely many semidegrees \(\delta _1, \ldots , \delta _k\) such that for all \(f \in A{\setminus }\{0\}\),

$$\begin{aligned} \delta (f) = \max \{\delta _1(f), \ldots , \delta _k(f)\} \end{aligned}$$
(14)

Remark B.2

If \(\delta \) is integer-valued on \(A{\setminus } \{0\}\) (i.e. \(\delta (f) = -\infty \) iff \(f = 0\)), then \(\delta \) is a semidegree iff \(-\delta \) is a discrete valuation.

Let \(A \subseteq B\) be \(\mathbb {K}\)-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.

Lemma B.3

Let \(\delta _1, \ldots , \delta _m\) be semidegrees on A which are integer-valued on \(A{\setminus }\{0\}\), and \(\delta := \max \{\delta _1, \ldots , \delta _m\}\). For each i, \(1 \le i \le m\), let \(\eta _{ij}\), \(1 \le j \le m_i\), be the extension of \(\delta _i\) to B. Define \(\eta := \max \{\eta _{ij}, 1 \le i \le m, 1 \le j \le m_i\}\). If A is integrally closed, then \(B^\eta \) is integral over \(A^\delta \). If in addition B is integrally closed, then \(B^\eta \) is the integral closure of \(A^\delta \) in the quotient field of \(B^\eta \).

Proof

By construction, the restriction of \(\eta \) to A is precisely \(\delta \), so that \(A^\delta \subseteq B^\eta \). The last assertion of the lemma follows from the first by Lemma 4.4. We now demonstrate the first assertion. Let t be an indeterminate. Identify \(B^\eta \) with a subring of B[t] as in (2). Let \(f \in B {\setminus } \{0\}\) and \(d' := \eta (f)\). It suffices to show that \(ft^{d'} \in B^\eta \) satisfies an integral equation over \(A^\delta \). Let the minimal polynomial of f over K be

$$\begin{aligned} P(T) := T^d + \sum _e g_e T^{d-e} \end{aligned}$$
(15)

and \(L'\) be the Galois closure of L over K. Since \(L'\) is Galois over K, it contains all the roots \(f_1, \ldots , f_d\) of P(T). Since L / K is finite and separable, each \(f_i = \sigma _i(f)\) for some \(\sigma _i \in {{\mathrm{Gal}}}(L'/K)\). For each ij, \(1 \le i \le m\) and \(1 \le j \le m_i\), let \(\{\eta '_{ijk}: 1 \le k \le l_{ij}\}\) be the extensions of \(\eta _{ij}\) to \(L'\). Define

$$\begin{aligned} \eta '_i&:= \max \{\eta '_{ijk}: 1 \le j \le m_i,\ 1 \le k \le l_{ij}\},\ 1 \le i \le m,\quad \text {and}\\ \eta '&:= \max \{\eta '_i : 1 \le i \le m\}. \end{aligned}$$

Since each of \(\delta _i\), \(\eta _{ij}\) and \(\eta '_{ijk}\)’s is the negative of a discrete valuation, it follows that each \(\eta '_{ijk} = \eta '_{i11} \circ \sigma _{ijk}\) for some \(\sigma _{ijk} \in {{\mathrm{Gal}}}(L'/K)\) (Zariski and Samuel 1975, Theorem VI.12, Corollary 3). It follows that for all ij,

$$\begin{aligned} \eta '_i(f_j)&= \max \{\eta '_{i11} \circ (\sigma _{ij'k'} \circ \sigma _j)(f): 1 \le j' \le m_i,\ 1 \le k' \le l_{ij} \}\\&= \max \{\eta '_{ij''k''}(f): 1 \le j'' \le m_i,\ 1 \le k'' \le l_{ij} \}\\&= \eta '_i(f). \end{aligned}$$

Note that \(\eta '_i|_K = \delta _i\) for each i. Since each \(g_e\) (from (15)) is an e-th symmetric polynomial in \(f_1, \ldots , f_d\), it follows that for all i, \(1 \le i \le m\), and all e, \(1 \le e \le d\),

$$\begin{aligned} \delta _i(g_e) = \eta '_i(g_e) \le e\eta '_i(f). \end{aligned}$$
(16)

Since \(\eta '|_B = \eta \), it follows that \(d' = \eta (f) = \eta '(f)\). By definition of \(\eta '\), there exists i, \(1 \le i \le m\), such that \(d' = \eta '_i(f) \ge \eta '_{i'}(f)\) for all \(i'\), \(1 \le i' \le m\). It then follows from (16) that

$$\begin{aligned} \delta _i(g_e) \le ed'\quad \text {for all}\ i,\; 1 \le i \le m. \end{aligned}$$
(17)

Now recall that A is integrally closed, so that \(g_e \in A\) for all e (Atiyah and Macdonald 1969, Proposition 5.15). Since inequality (17) implies that \(\delta (g_e) \le ed'\), it follows that \(g_et^{ed'} \in A^\delta \) for all e. Consequently \(ft^{d'}\) satisfies the integral equation

$$\begin{aligned} \tilde{P}(T) := T^d + \sum _e g_e t^{ed'} T^{d-e} \end{aligned}$$

over \(A^\delta \). Therefore \(B^\eta \) is integral over \(A^\delta \), as required. \(\square \)

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Mondal, P. When is the Intersection of Two Finitely Generated Subalgebras of a Polynomial Ring Also Finitely Generated?. Arnold Math J. 3, 333–350 (2017). https://doi.org/10.1007/s40598-017-0068-8

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