Abstract Nonsense

Loci of Holomorphic Functions and the Inverse Function Theorem (Pt. III)

Point of Post: This is a continuation of this post.

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Thus, we have following theorem:

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Theorem: Let $U\subseteq\mathbb{C}^2$ be open and $f\in\mathcal{O}(U)$ be non-singular such that $Z(f)$ is connected. Then, $Z(f)$ is a Riemann surface.

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This is somewhat unsatisfactory because while the notion of non-singularity is easy to identify, it seems fairly difficult to prove that $Z(f)$ is connected. While this is true in general, there is at least one case where this is not so bad. In particular, let’s try to make our life easier. Instead of trying to find conditions for when $Z(f)$ is connected for general $f\in\mathcal{O}(U)$ let’s instead try to find conditions for when $Z(f)$ is connected when $f\in\mathbb{C}[w,v]\subseteq\mathcal{O}(U)$.

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To try to figure out what the appropriate conditions are for such $f$ we look at a typical example of when $Z(f)$ fails to be connected. For the sake of simplicity we are going to deal with $f\in\mathbb{R}[v,w]$ since we can’t picture $\mathbb{C}^2$, but the same idea goes through. Think about what happens if take the union of a parabola and a non-intersecting line–this is a locus which is clearly not connected. What went wrong? Where did these pieces come from? Well, drawing a random example of a non-intersecting parabola and line leads me to the parabola $v=w^2$ and the line $v=-(1+w)$. So, how do we describe their union as the locus of a single polynomial? Well, want want a polynomial $f(v,w)$ such that $f(v,w)=0$ if and only if one of $v=w^2$ or $v=-(1+w)$. Clearly, up to units, such a polynomial will be just $f(v,w)=(v+1+w)(v-w^2)$. We see then that  the disjoint pieces of $Z(f)$ came from the different factors of $f$. Indeed, one can check that if one writes down a random polynomial in $\mathbb{R}[v,w]$ then their locus will be disconnected (there is no rigorous backing for this, it’s just for illustrative purposes!).

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With the above analysis it seems like a good place to start looking for polynomials with irreducible loci are among the polynomials that can’t be factored. Indeed, recall that a polynomial $f\in\mathbb{C}[w,v]$ is irreducible if whenever $f=gh$ then either $g$ or $h$ is a constant. We  then have the following theorem:

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Theorem: Let $f\in\mathbb{C}[w,v]$ be irreducible. Then, $Z(f)$ is connected.

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Unfortunately, this is another theorem that, while not too difficult, would take us too far afield from our current goals. Really, this is a theorem of basic algebraic geometry and thus can be found in almost any good text on the subject. In particular, we shall refer an interested reader to [10] if they are interested in the proof of this theorem.

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Let’s look at, at least one example. Consider $f(w,v)=w^2+v^2-1$. We claim that $f$ is irreducible and non-singular. Non-singularity is obvious because if $f_w(w,v)=f_v(w,v)=0$ then $(w,v)=(0,0)$ but $(0,0)\notin Z(f)$. To see that $f(w,v)$ is irreducible one can take the elementary approach by just assuming that $f$ factors into two polynomials and showing that one must be a unit. This, while tedious, is fairly straightforward and left to the reader (and this linear system solving devices).

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References:

[1] Varolin, Dror. Riemann Surfaces by Way of Complex Analytic Geometry. Providence, RI: American Mathematical Society, 2011. Print.

[2] Miranda, Rick. Algebraic Curves and Riemann Surfaces. Providence, RI: American Mathematical Society, 1995. Print.

[3] Forster, Otto. Lectures on Riemann Surfaces. New York: Springer-Verlag, 1981. Print.

[4] Conway, John B. Functions of One Complex Variable. New York: Springer-Verlag, 1978. Print.

[5] Gong, Sheng. Concise Complex Analysis. Singapore: World Scientific, 2001. Print

[6] Hörmander, Lars. An Introduction to Complex Analysis in Several Variables. Princeton, NJ: Van Nostrand, 1966. Print.

[7] Krantz, Steven G. Function Theory of Several Complex Variables. New York: Wiley, 1982. Print.

[8] Ohsawa, T. Analysis of Several Complex Variables. Providence, RI: American Mathematical Society, 2002. Print.

[9] Ebeling, Wolfgang. Functions of Several Complex Variables and Their Singularities. Providence, RI: American Mathematical Society, 2007. Print.

[10] Shafarevich, I. R. Basic Algebraic Geometry. Berlin: Springer-Verlag, 1994. Print.

October 3, 2012 -