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Let $\varphi_1(u,v)$ and $\varphi_2(u,v)$ be two entire or meromorphic functions in the two complex variables $u$ and $v$. If they are both polynomials, then Bezout's Theorem says that the set of common roots $(u,v)$ of $\varphi_1(u,v)=0$ and $\varphi_2(u,v)=0$ is, with suitable qualifications, a finite set whose cardinality is the product of the degrees of $\varphi_1(u,v)$ and $\varphi_2(u,v)$. There is a similar statement for the common roots of three polynomial equations in three variables, and so on.

Here is my question: Is there a corresponding theorem for the case when $\varphi_1(u,v)$ and $\varphi_2(u,v)$ are transcendental functions, i.e., in general is their common set of roots an infinite discrete point set in two-dimensional complex space?

If one is transcendental and the other is linear, then the result is true...take the case of $v=\sin u$ and $v=0$. Picard's theorem gives a general answer in that case. But, what happens when both functions are transcendental?

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  • $\begingroup$ It seems the question is how to count `intersection at infinity' (which you have to do in the Bezout Theorem), otherwise it is easy to give counterexamples... $\endgroup$
    – t3suji
    Commented Feb 4, 2010 at 14:01
  • $\begingroup$ Everyone seems to be misreading the question. We aren't asking that the intersection set be finite, let alone that we can give a formula for its size. We just want to know that it is discrete. $\endgroup$
    – David E Speyer
    Commented Feb 4, 2010 at 15:46
  • $\begingroup$ @David Speyer: Well, if that was the intended meaning, it does not have much to do with Bezout's theorem --- it is a claim about structure of analytic sets (which probably follows from Weierstrass Preparation Lemma). Can the OP confirm that this is the question? Because as I read it, 'infinite discrete set' means just that: a discrete set that fails to be finite. $\endgroup$
    – t3suji
    Commented Feb 4, 2010 at 15:59
  • $\begingroup$ @t3suji Is your point that sometimes the intersection is actually a finite discrete set, as in (y+x e^x, x+y+xe^x)? That's certainly true. I guess we need the OP to confirm whether we were supposed to focus on discreteness or infinitude. $\endgroup$
    – David E Speyer
    Commented Feb 4, 2010 at 16:28
  • $\begingroup$ @David Speyer: yes, that's what I meant --- that if intersection is a finite discrete set, counting points at infinity of projective plane makes it infinite... See also the answer of Steven Gubkin. $\endgroup$
    – t3suji
    Commented Feb 4, 2010 at 16:33

3 Answers 3

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I am sure that the theorem you want is true, but I am missing one technical reference.

Claimed Theorem: Let $f(x,y)$ and $g(x,y)$ be entire functions on $\mathbb{C}^2$. Then either $\{ f=0 \}$ and $\{ g=0 \}$ have a one dimensional overlap, or $\{ f=g=0 \}$ is discrete.

"Proof": If $\{ f=g=0 \}$ is not discrete, then it has an accumulation point $z$. Suppose $\{ f=0 \}$ is smooth at $z$. Then, by the implicit function theorem, we can locally parameterize $\{ f=0 \}$ as $\{ (a(t), b(t)): t \in D \}$ where $D$ is a small disc and $a$ and $b$ are alanlytic functions on $D$, with $(a(0), b(0))=z$. Then the function $g(a(t), b(t))$ has infinitely many zeroes with an accumulation point at $t=0$, so it must be identically zero. Thus, $g$ vanishes on an open set in $\{f=0 \}$.` QED

Now, the gap in the above is that $z$ might not be a smooth point.

What I am quite certain is true, but I don't know a reference for, is that we have resolution of singularities for analytic germs the same way we do for polynomials. So, for any nonzero analytic function $f(x,y)$ and any point $z$, there is a neighborhood of $z$ where we can factor $f$ as $f_1 f_2 \cdots f_r$, each an analytic function, and such that $\{ f_i=0 \}$, near $z$, can be paratemerized $\{ (a_i(t), b_i(t)): t \in D \}$. One can probably prove this by brute force, but I'm sure one of our readers knows a reference.

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  • $\begingroup$ Does your last assertion by any chance have something to do with Puiseux series? I ask because they came up in my class as an example of a non-discretely valued field, and some of my students asked me what they were "for"...(P.S.: I believe your claimed theorem.) $\endgroup$
    – Pete L. Clark
    Commented Feb 4, 2010 at 16:05
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    $\begingroup$ Part (6) of eom.springer.de/a/a012410.htm summarizes properties of complex analytic sets, including generalization of Claimed Theorem; it has references. $\endgroup$
    – t3suji
    Commented Feb 4, 2010 at 16:07
  • $\begingroup$ @Pete, Puiseux series are quite useful when proving desingularizition for curves---the argument is due to Newton, if I recall correctly. $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Feb 4, 2010 at 17:26
  • $\begingroup$ Your "claimed theorem" is a nice answer. What would the theorem say for 3 functions in 3 variables, etc? $\endgroup$
    – Mark B Villarino
    Commented Feb 4, 2010 at 20:57
  • $\begingroup$ @Pete: Puiseux series are also very important examples in Model Theory. In the real case, they are the setting for the use of infinitesimals (see e.g. the book Real Algebraic Geometry by Bochnak, Coste and Roy). $\endgroup$
    – Thierry Zell
    Commented Aug 14, 2010 at 15:41
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sin(x) and x + sin(x) are both transcendental right?

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I am not an expert but I'd like to add that there is a theory called Value Distribution Theory (or Nevanlinna Theory) which counts how many elements there are in your set within a ball of radius $R$ and the behaviour as $R \to \infty$. In particular, something like the "second main theorem with moving targets" should give what you want at least when your analytic curves $\phi = 0$ are graphs.

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