User mark b villarino - MathOverflow

Is a function which is finitely multiple-valued in each variable separately, also finitely multiple-valued in all its variables jointly?

2010-02-11T15:36:04Z

It is well known that under suitable conditions, a function which is:

a polynomial in each variable separately is a polynomial in all its variables jointly.
a rational function in each variable separately is a rational function.
a holomorphic function in each variable separately is holomorphic in all its variables.

A complete analytic function can be single-valued or multiple-valued according as it does not have, or does have, branch points. The algebraic functions are examples of the latter.

Here is my question: is a complete analytic function, which is finitely multiple-valued in each variable separately, also finitely multiple-valued jointly?

Routh's theorem in three dimensions

2010-03-14T16:39:38Z

The most well known case of Routh's triangle theorem is: If the sides BC, CA,and AB are trisected at the points D, E, and F, respectively, then the area of the inside triangle formed by AD, BE, CF is $\dfrac{1}{7}$th of the area of that of the triangle ABC.

Here is my question: can Routh's theorem be generalized to a tetrahedron which is cut by 4 planes through its 4 vertices and cutting the opposite faces appropriately?

As far as I know, this question has never been contemplated in the literature.

Answer by Mark B Villarino for Value of "of course" in the mathematical literature

2010-02-25T06:16:55Z

Many years ago, a professor of mine in a graduate algebra course wrote down a totally impenetrable statement and then added:

"Of course, it's obvious. It may not be clear that it's obvious...but it's obvious."

The words "it's obvious", etc., should be discarded just as the words "We have..." should be discarded. Also, the words "Consider the following function...." As Estermann comented..."I do not know what that means."

Is there a "Bezout's theorem" for analytic curves?

2010-02-04T13:43:47Z

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?

the group law for an elliptic curve

2010-02-03T23:05:28Z

Let $\varphi(u)$ be holomorphic in the neighborhood of the origin of the complex plane. One says that $\varphi(u)$ admits an algebraic addition theorem if it satisfies a functional equation of the form $G[\varphi(u). \varphi(v), \varphi(u+v)]=0,$ where where G(X,Y,Z) is a non vanishing polynomial in the three variables X,Y,Z with complex constant coefficients, while $u,v, u+v$ are in the domain of $\varphi(u)$. Examples are the rational functions, the exponential function, the trigonometric functions, and the elliptic functions. Then it can be proved that $\varphi(u)$ and $\varphi'(u)$ are connected by an algebraic equation $A[\varphi(u)$,$\varphi'(u)]=0$, which defines the ELLIPTIC CURVE parameterized by $\varphi(u)$. It is known that ANY elliptic curve can be realized in this way.

If one finds the greatest common divisor of G(X,Y,Z) and $X'\frac{dG}{d Y}-Y'\frac{dG}{d X},$, where $X'$ means the derivative with respect to $u$, etc., we obtain an irreducible polynomial $D(Z,X,X',Y,Y')$, which, when put equal to zero, gives the GROUP LAW for A(X,X')=0. If the degree of D in Z is equal to one, then the group law is rational in X,X',Y,Y'. Here is the question:

Prove: the degree of D in Z is one iff $\varphi(u)$ is uniform (i.e., has no branch points)

The proof must not use the properties of $\varphi(u)$ as rational, trigonometric, or elliptic functions...rather, only the uniformity of $\varphi(u)$.

I have not seen a proof...it is desirable that one be found. There ARE proofs using the properties of the rational, trigonometric, and elliptic functions.

This gives an elementary algorithm for finding the group law for any elliptic curve without detouring through the Weierstrass $\wp$ function.

Answer by Mark B Villarino for addition-theorem polynomials

2010-02-03T22:34:16Z

If X' means the derivative with respect to u and Y' that wrsp y, etc., then one condition is: Elimination of Z between G=0 and $X'\frac{\partial G}{\partial Y}=Y'\frac{\partial G}{\partial X}$ leads to only a single equation between X and X' for all values of Y and Y' (see Forsyth, page 357) – Mark B Villarino 0 secs ago

Answer by Mark B Villarino for addition-theorem polynomials

2010-02-02T13:51:49Z

It is a famous theorem of Weierstrass that the only meromorphic functions admitting an algebraic addition theorem are rational functions, or rational functions of the exponential function, or elliptic functions. What has NOT been answered is: given a polynomial G(Z,X,Y), in the three variables X,Y,Z, is it an addition-theorem polynomial? Which formal characteristics of G characterize it as such a polynomial? As far as I know, this question has never been investigated.

Comment by Mark B Villarino

2010-03-15T11:18:43Z

The main problem is HOW do you define a "sensible cut"?...Do you "trisect" the area of the opposite face, do any three of the cutting planes define a corresponding 1/7 triangle in a face of the tetrahedon? Yes, T(A)=B, but I am looking for an explicit construction.

Comment by Mark B Villarino

2010-02-11T17:52:55Z

I do not see how you can reach an infinite number of values for $f(x_0,y_0)$ by moving both coordinates together, if you ALWAYS obtain only a finite number of values by varying only one of them

Comment by Mark B Villarino

2010-02-11T17:13:02Z

no....it is possible that for each different y the number of values of f changes...the question asks if there is a uniform upper bound for the number of values of f regardless of the y...and vice versa

Comment by Mark B Villarino

2010-02-11T17:01:07Z

x+y is single-valued as a function of (x,y)...my question asks if for each $(x,y_0)$, $f(x,y_0)\equiv g(x)$ is finitely multiple-valued as a function of $x$, and $f(x_0,y)\equiv h(y)$ is a finitely multiple-valued function of $y$, then is $f(x,y)$ a finitely multiple-valued function of the pair $(x,y)$? If, for example, one writes "algebraic" in place of "finitely multiple-valued", then under certain conditions the joint function is algebraic.

Comment by Mark B Villarino

2010-02-04T20:57:48Z

Your "claimed theorem" is a nice answer. What would the theorem say for 3 functions in 3 variables, etc?

Comment by Mark B Villarino

2010-02-04T06:23:23Z

Yes, the function $\varphi$.

Comment by Mark B Villarino

2010-02-03T23:14:25Z

the domain of $\varphi(u)$ is initially a neighborhood of the origin..indeed, the group law allows one to continue it to all of the complex plain, if the degree of D is one...

Comment by Mark B Villarino

2010-02-03T22:33:11Z

Comment by Mark B Villarino

2010-02-02T19:07:11Z

The specific reason for my interest is that the problem is simply stated, concrete, interesting in itself, and unanswered. Moreover, I am preparing an expository paper on this beautiful classical theory since the presentations in Hancock, and in Forsyth, have definite mistakes and errors, quite apart from being misleading and difuse. Koebe (in his thesis) and Forsyth prove certain properties of G, for example the formula for its degree in Z, but leave the question, there. Since a century has passed, one would hope that fresh insights might lead to further results. This website is ideal.