Lüroth’s theorem (a “constructive” proof)

Lüroth’s theorem (Lüroth 1876 for $$k = \mathbb{C}$$, Steinitz 1910 in general). If $$k \subseteq K$$ are fields such that $$k \subseteq K \subseteq k(x)$$, where $$x$$ is an indeterminate over $$k$$, then $$K = k(g)$$ for some rational function $$g$$ of $$x$$ over $$k$$.

I am going to present a “constructive” proof of Lüroth’s theorem due to Netto (1895) that I learned from Schinzel’s Selected Topics on Polynomials (and give some applications to criteria for proper polynomial parametrizations). The proof uses the following result which I am not going to prove here:

Proposition (with the set up of Lüroth’s theorem). $$K$$ is finitely generated over $$k$$, i.e. there are finitely many rational functions $$g_1, \ldots, g_s \in k(x)$$ such that $$K = k(g_1, \ldots, g_s)$$.

The proof is constructive in the following sense: given $$g_1, \ldots, g_s$$ as in the proposition, it gives an algorithm to determine $$g$$ such that $$K = k(g)$$. We use the following notation in the proof: given a rational function $$h \in k(x)$$, if $$h = h_1/h_2$$ with polynomials $$h_1, h_2 \in k[x]$$ with $$\gcd(h_1, h_2) = 1$$, then we define $$\deg_\max(h) := \max\{\deg(h_1), \deg(h_2)\}$$.

Proof of Lüroth’s theorem

It suffices to consider the case that $$K \neq k$$. Pick $$g_1, \ldots, g_s$$ as in the proposition. Write $$g_i = F_i/G_i$$, where

• $$\gcd(F_i, G_i) = 1$$ (Property 1).

Without loss of generality (i.e. discarding $$g_i \in k$$ or replacing $$g_i$$ by $$1/(g_i + a_i)$$ for appropriate $$a_i \in k$$ if necessary) we can also ensure that

• $$\deg(F_i) > 0$$ and $$\deg(F_i) > \deg(G_i)$$ (Property 2).

Consider the polynomials $H_i := F_i(t) – g_iG_i(t) \in K[t] \subset k(x)[t], i = 1, \ldots, s,$ where $$t$$ is a new indeterminate. Let $$H$$ be the greatest common divisor of $$H_1, \ldots, H_s$$ in $$k(x)[t]$$ which is also monic in $$t$$. Since the Euclidean algorithm for computing $$\gcd$$ respects the field of definition, it follows that:

• $$H$$ is also the greatest common divisor of $$H_1, \ldots, H_s$$ in $$K[t]$$, which means, if $$H = \sum_j h_j t^j$$, then each $$h_j \in K$$ (Property 3).

Let $$H^* \in k[x,t]$$ be the polynomial obtained by “clearing the denominator” of $$H$$; in other words, $$H = H^*/h(x)$$ for some polynomial $$h \in k[x]$$ and $$H^*$$ is primitive as a polynomial in $$t$$ (i.e. the greatest common divisor in $$k[x]$$ of the coefficients in $$H^*$$ of powers of $$t$$ is 1). By Gauss’s lemma, $$H^*$$ divides $$H^*_i := F_i(t)G_i(x) – F_i(x)G_i(t)$$ in $$k[x,t]$$, i.e. there is $$Q_i \in k[x,t]$$ such that $$H^*_i = H^* Q_i$$.

Claim 1. If $$\deg_t(H^*) < \deg_t(H^*_i)$$, then $$\deg_x(Q_i) > 0$$.

Proof of Claim 1. Assume $$\deg_t(H^*) < \deg_t(H^*_i)$$. Then $$\deg_t(Q_i) > 1$$. If in addition $$\deg_x(Q_i) = 0$$, then we can write $$Q_i(t)$$ for $$Q_i$$. Let $$F_i(t) \equiv \tilde F_i(t) \mod Q_i(t)$$ and $$G_i(t) \equiv \tilde G_i(t) \mod Q_i(t)$$ with $$\deg(\tilde F_i) < \deg(Q_i)$$ and $$\deg(\tilde G_i) < \deg(Q_i)$$. Then $$\tilde F_i(t)G_i(x) – F_i(x) \tilde G_i(t) \equiv 0 \mod Q_i(t)$$. Comparing degrees in $$t$$, we have $$\tilde F_i(t)G_i(x) = F_i(x) \tilde G_i(t)$$. It is straightforward to check that this contradicts Propeties1 and 2 above, and completes the proof of Claim 1.

Let $$m := \min\{\deg_\max(g_i): i = 1, \ldots, s\}$$, and pick $$i$$ such that $$\deg_\max(g_i) = m$$. Property 2 above implies that $$\deg_t(H^*_i) = \deg_x(H^*_i) = m$$. If $$\deg_t(H^*) < m$$, then Claim 1 implies that $$\deg_x(H^*) < \deg_x(H^*_i) = m$$. If the $$h_j$$ are as in Property 3 above, it follows that $$\deg_\max(h_j) < m$$ for each $$j$$. Since $$H^* \not\in k[t]$$ (e.g. since $$t-x$$ divides each $$H_i$$), there must be at least one $$h_j \not \in k$$. Since adding that $$h_j$$ to the list of the $$g_i$$ decreases the value of $$m$$, it follows that the following algorithm must stop:

Algorithm

• Step 1: Pick $$g_i := F_i/G_i$$, $$i = 1, \ldots, s$$, satisfying properties 1 and 2 above.
• Step 2: Compute the monic (with respect to $$t$$) $$\gcd$$ of $$F_i(t) – g_i G_i(t)$$, $$i = 1, \ldots, s$$, in $$k(x)[t]$$; call it $$H$$.
• Step 3: Write $$H = \sum_j h_j(x) t^j$$. Then each $$h_j \in k(g_1, \ldots, g_s)$$. If $$\deg_t(H) < \min\{\deg_\max(g_i): i = 1, \ldots, s\}$$, then adjoin all (or, at least one) of the $$h_j$$ such that $$h_j \not\in k$$ to the list of the $$g_i$$ (possibly after an appropriate transformation to ensure Property 2), and repeat.

After the last step of the algorithm, $$H$$ must be one of the $$H_i$$, in other words, there is $$\nu$$ such that $\gcd(F_i(t) – g_i G_i(t): i = 1, \ldots, s) = F_{\nu}(t) – g_{\nu}G_{\nu}(t).$

Claim 2. $$K = k(g_{\nu})$$.

Proof of Claim 2 (and last step of the proof of Lüroth’s theorem). For a given $$i$$, polynomial division in $$k(g_\nu)[t]$$ gives $$P, Q \in k(g_\nu)[t]$$ such that $F_i(t) = (F_{\nu}(t) – g_{\nu}G_{\nu}(t))P + Q,$ where $$\deg_t(Q) < \deg_t(F_{\nu}(t) – g_{\nu}G_{\nu}(t))$$. If $$Q = 0$$, then $$F_i(t) = (F_{\nu}(t) – g_{\nu}G_{\nu}(t))P$$, and clearing out the denominator (with respect to $$k[g_\nu]$$) of $$P$$ gives an identity of the form $$F_i(t)p(g_\nu) = (F_{\nu}(t) – g_{\nu}G_{\nu}(t))P^* \in k[g_\nu, t]$$ which is impossible, since $$F_{\nu}(t) – g_{\nu}G_{\nu}(t)$$ does not factor in $$k[g_\nu, t]$$. Therefore $$Q \neq 0$$. Similarly, $G_i(t) = (F_{\nu}(t) – g_{\nu}G_{\nu}(t))R + S,$ where $$R, S \in k(g_\nu)[t]$$, $$S \neq 0$$, and $$\deg_t(S) < \deg_t(F_{\nu}(t) – g_{\nu}G_{\nu}(t))$$. It follows that $F_i(t) – g_iG_i(t) = (F_{\nu}(t) – g_{\nu}G_{\nu}(t))(P – g_iR) + Q – g_iS.$ Since $$F_{\nu}(t) – g_{\nu}G_{\nu}(t)$$ divides $$F_{i}(t) – g_{i}G_{i}(t)$$ in $$k(x)[t]$$ and since $$\deg_t(Q – g_iS) < \deg_t(F_{\nu}(t) – g_{\nu}G_{\nu}(t))$$, it follows that $$Q = g_iS$$. Taking the leading coefficients (with respect to $$t$$) $$q_0, s_0 \in k(g_\nu)$$ of $$Q$$ and $$S$$ gives that $$g_i = q_0/s_0 \in k(g_\nu)$$, as required to complete the proof.

Applications

The following question seems to be interesting (geometrically, it asks when a given polynomial parametrization of a rational affine plane curve is proper).

Question 1. Let $$k$$ be a field and $$x$$ be an indeterminate over $$k$$ and $$g_1, g_2 \in k[x]$$. When is $$k(g_1, g_2) = k(x)$$?

We now give a sufficient condition for the equality in Question 1. Note that the proof is elementary: it does not use Lüroth’s theorem, only follows the steps of the above proof in a special case.

Corollary 1. In the set up of Question 1, let $$d_i := \deg(g_i)$$, $$i = 1, 2$$. If the $$\gcd$$ of $$x^{d_1} – 1, x^{d_2} – 1$$ in $$k[x]$$ is $$x – 1$$, then $k(g_1, g_2) = k(t)$. In particular, if $$d_1, d_2$$ are relatively prime and the characteristic of $$k$$ is either zero or greater than both $$d_1, d_2$$, then $k(g_1, g_2) = k(x)$.

Remark. Corollary 1 is true without the restriction on characteristics, i.e. the following holds: “if $$d_1, d_2$$ are relatively prime, then $k(g_1, g_2) = k(x)$.” François Brunault (in a comment to one of my questions on MathOverflow) provided the following simple one line proof: $$[k(x): k(g_1, g_2)]$$ divides both $$[k(x): k(g_i)] = d_i$$, and therefore must be $$1$$.

My original proof of Corollary 1. Following the algorithm from the above proof of Lüroth’s theorem, let $$H_i := g_i(t) – g_i(x)$$, $$i = 1, 2$$, and $$H \in k(x)[t]$$ be the monic (with respect to $$t$$) greatest common divisor of $$H_1, H_2$$.

Claim 1.1. $$H = t – x$$.

Proof. It is clear that $$t-x$$ divides $$H$$ in $$k(x)[t]$$, so that $$H(x,t) = (t-x)h_1(x,t)/h_2(x)$$ for some $$h_1(x,t) \in k[x,t]$$ and $$h_2(x) \in k[x]$$. It follows that there is $$Q_i(x,t) \in k[x,t]$$ and $$P_i(x) \in k[x]$$ such that $$H_i(x,t)P_i(x)h_2(x) = (t-x)h_1(x,t)Q_i(x,t)$$. Since $$h_2(x)$$ and $$(t-x)h_1(x,t)$$ have no common factor, it follows that $$h_2(x)$$ divides $$Q_i(x,t)$$, and after cancelling $$h_2(x)$$ from both sides, one can write $H_i(x,t)P_i(x) = (t-x)h_1(x,t)Q’_i(x,t),\ i = 1, 2.$ Taking the leading form of both sides with respect to the usual degree on $$k[x,t]$$, we have that $(t^{d_i} – x^{d_i})x^{p_i} = a_i(t-x)\mathrm{ld}(h_1)\mathrm{ld}(Q’_i)$ where $$a_i \in k \setminus \{0\}$$ and $$\mathrm{ld}(\cdot)$$ is the leading form with respect to the usual degree on $$k[x,t]$$. Since $$\gcd(x^{d_1} – 1, x^{d_2} – 1) = x – 1$$, it follows that $$\mathrm{ld}(h_1)$$ does not have any factor common with $$t^{d_i} – x^{d_i}$$, and consequently, $$t^{d_i} – x^{d_i}$$ divides $$(t-x)\mathrm{ld}(Q’_i)$$. In particular, $$\deg_t(Q’_i) = d_i – 1$$. But then $$\deg_t(h_1) = 0$$. Since $$H = (t-x)h_1(x)/h_2(x)$$ is monic in $$t$$, it follows that $$H = t – x$$, which proves Claim 1.1.

Since both $$H_i$$ are elements of $$k(g_1, g_2)[t]$$, and since the Euclidean algorithm to compute $$\gcd$$ of polynomials (in a single variable over a field) preserves the field of definition, it follows that $$H \in k(g_1, g_2)[t]$$ as well (this is precisely the observation of Property 3 from the above proof of Lüroth’s theorem). Consequently $$x \in k(g_1, g_2)$$, as required to prove Corollary 1.

References

• Andrzej Schinzel, Selected Topics on Polynomials, The University of Michigan Press, 1982.

Publications

Overview

Starting from my PhD thesis Towards a Bezout-type Theory of Affine Varieties written at University of Toronto under the supervision of Pierre Milman, my research in math falls under two broad themes:

• Compactification of affine varieties
• Affine Bézout problem

I am in particular interested in one of the simplest cases of the first problem:

• Compactification of $\mathbb{C}^2$

Papers

Compactification of $\mathbb{C}^2$

Preprints

• Normal equivariant compactifications of $\mathbb{G}^2_a$ of Picard rank one, arXiv:1610.03563.
• Mori dream surfaces associated with curves with one place at infinity, arXiv:1312.2168.
• Analytic compactifications of $\mathbb{C}^2$ part II – one irreducible curve at infinity, arXiv:1307.5577.

Affine Bézout problem

Preprints

• Intersection multiplicity, Milnor number and Bernstein’s theorem, arXiv:1607.04860

Simplest singularity on non-algebraic normal Moishezon surfaces

A classical question in complex analytic geometry is to understand when a given analytic space is algebraic (i.e. analytiﬁcation of an algebraic scheme). A necessary condition for this to hold is that the transcendence degree of the ﬁeld of global meromorphic functions must be equal to the dimension of the space, i.e. the space has to be Moishezon. For dimension 2, it is a classical result that it is also suﬃcient, provided the space is nonsingular (Chow and Kodaira, 1952).

In general it is not clear how to determine algebraicity of normal (singular) Moishezon surfaces and our understanding of non-algebraic Moishezon surfaces, more precisely what prevents them from being algebraic, remains incomplete (Schröer [Sch00] gives a necessary and suﬃcient for algebraicity, but it is not very suitable for computation in a given case). We [Mon16] gave an example of a non-algebraic normal Moishezon surface $$X$$ which has the simplest possible singularity in the following sense: $$X$$ has only one singular point $$P$$, and the singularity at $$P$$

1. has multiplicity 2 and geometric genus 1,
2. is almost rational in the sense of [Ném07], and
3. is a Gorenstein hypersurface singularity which is minimally elliptic (in the sense of [Lau77]).

The claim that singularity of $$X$$ is the simplest possible is based on combining the preceding facts with the following observations:

• a Moishezon surface whose singularities are rational (i.e. with geometric genus zero) is algebraic [Art62], and
• minimally elliptic Gorenstein singularities form in a sense the simplest class of non-rational singularities.

The weighted dual graph of the resolution of singularity of $$X$$ at $$P$$ is of type $$D_{9,∗,0}$$ in the classiﬁcation of [Lau77] and the self-intersection number of its fundamental divisor is −2. It follows from [Lau77, Table 2] that the singularity at the origin of $$z^2 = x^5 + xy^5$$ is also of the same type.

References

• [Art62] Michael Artin, Some numerical criteria for contractability of curves on algebraic surfaces, Amer. J. Math., 84:485–496,
• [Lau77] Henry B. Laufer, On minimally elliptic singularities, Amer. J. Math., 99(6):1257–1295, 1977.
• [Mon16] Pinaki Mondal, Algebraicity of normal analytic compactiﬁcations of $$\mathbb{C}^2$$ with one irreducible curve at inﬁnity, Algebra & Number Theory, 10(8), 2016.
• [Ném07] András Némethi, Graded roots and singularities, In Singularities in geometry and topology, pages 394–463. World Sci. Publ., Hackensack, NJ, 2007.
• [Sch00] Stefan Schröer, On contractible curves on normal surfaces, J. Reine Angew. Math., 524:1–15, 2000.