User mark b villarino - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T15:51:33Z http://mathoverflow.net/feeds/user/3725 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/14998/is-a-function-which-is-finitely-multiple-valued-in-each-variable-separately-also Is a function which is finitely multiple-valued in each variable separately, also finitely multiple-valued in all its variables jointly? Mark B Villarino 2010-02-11T15:36:04Z 2010-04-17T05:22:17Z <p>It is well known that under suitable conditions, a function which is:</p> <ol> <li><p>a polynomial in each variable separately is a polynomial in all its variables jointly.</p></li> <li><p>a rational function in each variable separately is a rational function.</p></li> <li><p>a holomorphic function in each variable separately is holomorphic in all its variables.</p></li> </ol> <p>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.</p> <p>Here is my question: <strong><em>is a complete analytic function, which is finitely multiple-valued in each variable separately, also finitely multiple-valued jointly?</em></strong></p> http://mathoverflow.net/questions/18179/rouths-theorem-in-three-dimensions Routh's theorem in three dimensions Mark B Villarino 2010-03-14T16:39:38Z 2010-03-14T18:49:53Z <p>The most well known case of Routh's triangle theorem is: <em>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.</em></p> <p>Here is my question: <strong><em>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?</em></strong></p> <p>As far as I know, this question has never been contemplated in the literature.</p> http://mathoverflow.net/questions/16193/value-of-of-course-in-the-mathematical-literature/16372#16372 Answer by Mark B Villarino for Value of "of course" in the mathematical literature Mark B Villarino 2010-02-25T06:16:55Z 2010-02-25T06:16:55Z <p>Many years ago, a professor of mine in a graduate algebra course wrote down a totally impenetrable statement and then added:</p> <p>"Of course, it's obvious. It may not be <em>clear</em> that it's obvious...but it's obvious."</p> <p>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."</p> http://mathoverflow.net/questions/14141/is-there-a-bezouts-theorem-for-analytic-curves Is there a "Bezout's theorem" for analytic curves? Mark B Villarino 2010-02-04T13:43:47Z 2010-02-04T17:17:18Z <p>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 <strong><em>Bezout's Theorem</em></strong> says that the set of <strong><em>common roots</em></strong> $(u,v)$ 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.</p> <p>Here is my question: Is there a corresponding theorem for the case when $\varphi_1(u,v)$ and $\varphi_2(u,v)$ are <strong><em>transcendental</em></strong> functions, i.e., in general <strong><em>is</em></strong> <strong><em>their common set of roots an infinite discrete point set in two-dimensional complex space?</em></strong></p> <p>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 <strong><em>both functions</em></strong> are transcendental?</p> http://mathoverflow.net/questions/14078/the-group-law-for-an-elliptic-curve the group law for an elliptic curve Mark B Villarino 2010-02-03T23:05:28Z 2010-02-04T05:24:39Z <p>Let $\varphi(u)$ be holomorphic in the neighborhood of the origin of the complex plane. One says that $\varphi(u)$ admits an <strong><em>algebraic addition theorem</em></strong> 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.</p> <p>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:</p> <p>Prove: <strong><em>the degree of D in Z is one iff $\varphi(u)$ is uniform</em></strong> (i.e., has no branch points)</p> <p>The proof must not use the properties of $\varphi(u)$ as rational, trigonometric, or elliptic functions...rather, only the uniformity of $\varphi(u)$.</p> <p>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.</p> <p>This gives an elementary algorithm for finding the group law for any elliptic curve without detouring through the Weierstrass $\wp$ function.</p> http://mathoverflow.net/questions/13741/addition-theorem-polynomials/14070#14070 Answer by Mark B Villarino for addition-theorem polynomials Mark B Villarino 2010-02-03T22:34:16Z 2010-02-03T22:34:16Z <p>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 </p> http://mathoverflow.net/questions/13741/addition-theorem-polynomials/13824#13824 Answer by Mark B Villarino for addition-theorem polynomials Mark B Villarino 2010-02-02T13:51:49Z 2010-02-02T13:51:49Z <p>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.</p> http://mathoverflow.net/questions/18179/rouths-theorem-in-three-dimensions/18187#18187 Comment by Mark B Villarino Mark B Villarino 2010-03-15T11:18:43Z 2010-03-15T11:18:43Z The main problem is HOW do you define a &quot;sensible cut&quot;?...Do you &quot;trisect&quot; 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. http://mathoverflow.net/questions/14998/is-a-function-which-is-finitely-multiple-valued-in-each-variable-separately-also Comment by Mark B Villarino Mark B Villarino 2010-02-11T17:52:55Z 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 http://mathoverflow.net/questions/14998/is-a-function-which-is-finitely-multiple-valued-in-each-variable-separately-also Comment by Mark B Villarino Mark B Villarino 2010-02-11T17:13:02Z 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 http://mathoverflow.net/questions/14998/is-a-function-which-is-finitely-multiple-valued-in-each-variable-separately-also/15002#15002 Comment by Mark B Villarino Mark B Villarino 2010-02-11T17:01:07Z 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 &quot;algebraic&quot; in place of &quot;finitely multiple-valued&quot;, then under certain conditions the joint function is algebraic. http://mathoverflow.net/questions/14141/is-there-a-bezouts-theorem-for-analytic-curves/14157#14157 Comment by Mark B Villarino Mark B Villarino 2010-02-04T20:57:48Z 2010-02-04T20:57:48Z Your &quot;claimed theorem&quot; is a nice answer. What would the theorem say for 3 functions in 3 variables, etc? http://mathoverflow.net/questions/14078/the-group-law-for-an-elliptic-curve Comment by Mark B Villarino Mark B Villarino 2010-02-04T06:23:23Z 2010-02-04T06:23:23Z Yes, the function $\varphi$. http://mathoverflow.net/questions/14078/the-group-law-for-an-elliptic-curve Comment by Mark B Villarino Mark B Villarino 2010-02-03T23:14:25Z 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... http://mathoverflow.net/questions/13741/addition-theorem-polynomials/13824#13824 Comment by Mark B Villarino Mark B Villarino 2010-02-03T22:33:11Z 2010-02-03T22:33:11Z 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) http://mathoverflow.net/questions/13741/addition-theorem-polynomials/13824#13824 Comment by Mark B Villarino Mark B Villarino 2010-02-02T19:07:11Z 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.