User adri&#225;n barquero - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T07:51:20Z http://mathoverflow.net/feeds/user/4170 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/102508/are-there-formulas-for-the-derivatives-zeta-fn0-of-dedekind-zeta-func Are there formulas for the derivatives $\zeta_{F}^{(n)}(0)$ of Dedekind zeta functions? Adrián Barquero 2012-07-18T01:55:51Z 2012-07-18T12:40:56Z <p>Let $F/\mathbb{Q}$ be a number field. I'm interested in knowing if there are formulas for the values of the derivatives $\zeta_{F}^{(n)}(0)$ of the Dedekind zeta function of $F$ at zero. </p> <p>Maybe if in the general case for an arbitrary number field there are no results, are there any results for particular types of number fields, like quadratic number fields or cyclotomic fields?</p> <p>I would also appreciate any references you can provide.</p> <p>Thank you for any help.</p> <p>PS: I would also be interested if anything is known only for the first values, say for $n = 1, 2, 3$ or so.</p> http://mathoverflow.net/questions/59605/reference-in-riemann-surfaces/59610#59610 Answer by Adrián Barquero for Reference in Riemann Surfaces Adrián Barquero 2011-03-25T21:02:33Z 2011-03-25T21:02:33Z <p>Frances Kirwan's book <a href="http://www.amazon.com/Complex-Algebraic-Mathematical-Society-Student/dp/0521423538/ref=sr_1_1?ie=UTF8&amp;qid=1301086873&amp;sr=8-1" rel="nofollow">Complex Algebraic Curves</a> has two really nice chapters on Riemann Surfaces and over all the level of the book is pretty decent to start with. The book is intended to be accessible to advanced undergraduates so perhaps not as advanced as you'd like, but it is a good reference nonetheless.</p> http://mathoverflow.net/questions/55594/what-is-known-about-the-conjectured-infinitude-of-regular-primes What is known about the conjectured infinitude of regular primes ? Adrián Barquero 2011-02-16T08:23:43Z 2011-02-16T08:55:24Z <p>I have read in some number theory books and in some online resources that it is known that there exist infinitely many irregular primes (a fact apparently proven quite some time ago, around 1915 by K. L. Jensen according to the Wikipedia entry). </p> <p>I haven't been able to find any reference, either in books or in the internet as to what the method for proving this might have been, but in any case, what I was more curious is about the conjectured existence of infinitely many <a href="http://en.wikipedia.org/wiki/Regular_prime" rel="nofollow">regular primes</a>.</p> <p>Even it is conjectured that there are "more" regular primes than irregular ones (about 61%), but the online references do not seem to say anything about its status, apart from saying that it is not known to be true. </p> <p>Thus the questions I have are:</p> <p>1) Are there any approaches at all to this problem?</p> <p>2) Are there any other conjectures known to imply the existence of infinitely many regular primes?</p> <p>3) Is it known why it is harder to prove this (other than the fact that it would give a proof of Fermat's Last Theorem for infinitely many prime exponents that does not involve heavy machinery =P) ?</p> <p>Thank you very much in advance.</p> http://mathoverflow.net/questions/31879/are-there-other-nice-math-books-close-to-the-style-of-tristan-needham/32210#32210 Answer by Adrián Barquero for Are there other nice math books close to the style of Tristan Needham? Adrián Barquero 2010-07-16T17:44:45Z 2010-07-16T17:44:45Z <p><a href="http://books.google.co.cr/books?id=L0d_Jhz07qsC&amp;printsec=frontcover&amp;dq=from+geometry+to+topology&amp;hl=es&amp;ei=y5dATJqcNIH78AbymeQU&amp;sa=X&amp;oi=book_result&amp;ct=result&amp;resnum=1&amp;ved=0CCoQ6AEwAA#v=onepage&amp;q&amp;f=false" rel="nofollow">From Geometry To Topology by H. Graham Flegg</a></p> <p>This book explains some basic topological concepts using a lot of examples and it has quite a lot of pictures. In fact, it is rather hard to find a single page that has no pictures in it. Very good for intuition indeed. And also very cheap since it is a Dover reprint.</p> http://mathoverflow.net/questions/28438/mathematics-and-autodidactism/28448#28448 Answer by Adrián Barquero for Mathematics and autodidactism Adrián Barquero 2010-06-17T00:22:01Z 2010-06-17T00:22:01Z <p>I have absolutely no idea of what learning process social interaction facilitates, but I can give you at least a really silly example of what sort of thing you do get when learning something from a person with more experience on the given subject. Everybody has seen at some point the basic expressions for the maximum and the minimum of two given real numbers $a, b \in \mathbb{R}$, say $max(a, b) = \dfrac{a + b + |a - b|}{2}$ and $min(a, b) = \dfrac{a + b - |a - b|}{2}$. I had always found it hard to remember the formulas. Well at one time I just realized that both formulas are completely obvious if you think of them as saying that for instance, if you want to get $max(a, b)$ then you just need to first step on the mid-point between $a, b$, which is $\dfrac{a + b}{2}$ and then you just have to "walk" from there half the distance between $a, b$ which is $\dfrac{|a - b|}{2}$ and similarly for the $min(a, b)$. </p> <p>So at some point the chance came that a friend of mine needed to use one of this formulas and he said, "Oh but I don't remember exactly how they were". So I explained to him what I just said and he told me that he had never seemed them that way and that now he was sure he will not forget them.</p> <p>My point is that this is exactly what you can get from an expert or from someone who has already thought about what you are learning at the moment, the hands on experience and the insight they have is what is most precious about this social interaction you refer to, it can give you the necessary ideas that you may not get from reading a book. You'll usually get more from this than what you get from a book or a paper. Obviously everything complements the other part so of course you can't expect to "be like Grothendieck" and learn everything directly from other people and you'll have to start taking the math from books, papers, wikipedia, etc. In a way I think of this as an aid in connecting the ideas I get from reading a book.</p> http://mathoverflow.net/questions/27749/what-are-some-correct-results-discovered-with-incorrect-or-no-proofs/27750#27750 Answer by Adrián Barquero for What are some correct results discovered with incorrect (or no) proofs? Adrián Barquero 2010-06-10T23:50:54Z 2010-06-10T23:50:54Z <p>I suppose we can cite here Fermat's Last Theorem as a prime example, although I'm not really sure about the connection between discovery and proof here. </p> http://mathoverflow.net/questions/8609/favorite-popular-math-book/27294#27294 Answer by Adrián Barquero for Favorite popular math book Adrián Barquero 2010-06-07T01:01:30Z 2010-06-07T01:01:30Z <p><strong>Title:</strong> Letters to a Young Mathematician</p> <p><strong>Author:</strong> Ian Stewart</p> <p><strong>Description:</strong> A beautiful book in which Stewart tries to convey in the form of letters from a mathematician to his grand daughter what sort of things does the profession of mathematics involves. It is very interesting since the letters advance from the time in which the grand daughter is in high school up until she is a professional mathematician doing research. I would recommend it without a doubt.</p> <p>From my own experience it has been really nice to come to this book at different times during the past years. I started college as an engineering student but on my third year I started taking courses from the mathematics program and I bought and read this book when I was just beginning. I had no real idea of what pure mathematics was all about (since I was used to the kind of calculus courses in which the emphasis is on computation rather than proving things) and this book gave me at least some perspective and a few hints of what may be ahead of me.</p> <p>Just for the record, I ended up switching my major to mathematics. </p> http://mathoverflow.net/questions/26497/maximal-ideals-in-the-ring-kx1-xn Maximal Ideals in the ring k[x1,...,xn ] Adrián Barquero 2010-05-30T20:25:24Z 2010-05-31T06:09:45Z <p>Hi. From one of the forms of Hilbert's Nullstellensatz we know that all the maximal ideals in a polynomial ring $k[x_1, \dots, x_n]$ where $k$ is an algebraically closed field, are of the form $(x_1 - a_1, \dots , x_n - a_n)$. So that any maximal ideal in this case is generated by polynomials $g_j \in k[x_1, \dots, x_n]$ for $j = 1, \dots , n$, where $g_j$ only depends on the variable $x_j$ (Obviously by taking $g_j = x_j - a_j$). Now, I'm interested in the case of a polynomial ring $k[x_1, \dots, x_n]$ where $k$ is an arbitrary field (i.e., I can't make use of the Nullstellensatz). I suppose this may no longer be the case, i.e., I don't expect any maximal ideal to be generated by n polynomials, each of them only dependent on one of the variables $x_j$, but my question is if maybe the maximal ideals can be generated by polynomials $g_j$ that only depend on the first $j$ variables $x_1, \dots , x_j$? If so, does anybody know how to prove this or can anyone suggest me some references that may help me?. Thanks.</p> http://mathoverflow.net/questions/25924/introducing-cryptology-to-undergraduates/25966#25966 Answer by Adrián Barquero for Introducing Cryptology to Undergraduates Adrián Barquero 2010-05-26T04:23:44Z 2010-05-26T04:23:44Z <p>You may want to consider taking a look at Simon Singh's book on the history of the subject "The Code Book" <a href="http://www.simonsingh.net/Crypto_Corner.html" rel="nofollow">link text</a></p> <p>He also has a downloadable application which has some very nice features, for example a working model of an enigma machine, etc. </p> <p>Also there's a really basic and short book by Annette Werner called "Elliptische Kurven in der Kryptographie" which may want to look at.</p> http://mathoverflow.net/questions/23643/books-about-history-of-recent-mathematics/23652#23652 Answer by Adrián Barquero for Books about history of recent mathematics Adrián Barquero 2010-05-06T01:02:36Z 2010-05-06T01:02:36Z <p>Another book which covers some recent history in its last chapters is The Queen of Mathematics: A Historically Motivated Guide to Number Theory. Also look at </p> <p>A History of Abstract Algebra by Israel Kleiner</p> <p>Episodes in the History of Modern Algebra (1800-1950)</p> <p>You do not have to restrict yourself to books, there also articles that touch on the history of a particular area of mathematics.</p> http://mathoverflow.net/questions/16087/defining-equations-for-secant-varieties/16092#16092 Answer by Adrián Barquero for defining equations for secant varieties Adrián Barquero 2010-02-22T21:40:26Z 2010-02-22T21:40:26Z <p>You could also try out Brendan Hassett's book "Introduction to Algebraic Geometry". In the fourth chapter he introduces secant varieties and even computes some examples using the techniques provided by Gröbner Bases. It is a really nice book.</p> http://mathoverflow.net/questions/102508/are-there-formulas-for-the-derivatives-zeta-fn0-of-dedekind-zeta-func/102521#102521 Comment by Adrián Barquero Adrián Barquero 2012-07-18T16:17:43Z 2012-07-18T16:17:43Z Thanks for the links Mrc. http://mathoverflow.net/questions/59770/are-the-non-trivial-zeros-of-zeta-simple/59774#59774 Comment by Adrián Barquero Adrián Barquero 2011-03-28T01:56:11Z 2011-03-28T01:56:11Z @GH Maybe it would be beaten by something like Zagier's title. Say something like &quot;A One Sentence Proof Of The Riemann Hypothesis&quot; =) http://mathoverflow.net/questions/35288/undergraduate-roadmap-to-algebraic-geometry/57196#57196 Comment by Adrián Barquero Adrián Barquero 2011-03-03T03:38:31Z 2011-03-03T03:38:31Z I seriously doubt that this is the best way to proceed. I mean, Hartshorne for a first approach to the subject, and at the undergraduate level? http://mathoverflow.net/questions/55594/what-is-known-about-the-conjectured-infinitude-of-regular-primes/55598#55598 Comment by Adrián Barquero Adrián Barquero 2011-02-16T15:16:08Z 2011-02-16T15:16:08Z Thanks for your answer Franz. http://mathoverflow.net/questions/55594/what-is-known-about-the-conjectured-infinitude-of-regular-primes Comment by Adrián Barquero Adrián Barquero 2011-02-16T15:10:28Z 2011-02-16T15:10:28Z Thanks a lot Kevin. http://mathoverflow.net/questions/54612/minimal-prerequisite-to-reading-wiles-proof-of-fermats-last-theorem/54776#54776 Comment by Adrián Barquero Adrián Barquero 2011-02-09T04:49:47Z 2011-02-09T04:49:47Z +1 These notes are really nice. Thanks. http://mathoverflow.net/questions/52979/integer-points-on-the-elliptic-curve-y2x317 Comment by Adrián Barquero Adrián Barquero 2011-01-24T04:40:11Z 2011-01-24T04:40:11Z @Gerry Myerson Thanks a lot, I'll try to see if the library has that volume with the collected papers and if not maybe I can try that interlibrary loan you mentioned. http://mathoverflow.net/questions/52979/integer-points-on-the-elliptic-curve-y2x317 Comment by Adrián Barquero Adrián Barquero 2011-01-23T20:41:27Z 2011-01-23T20:41:27Z @KConrad Silverman's book cites the following article by T. Nagell as a reference: Solution de quelque problemes dans la theorie arithmetique des cubiques planes du premier genre. Wid. Akad. Skrifter Oslo I, 1935. Nr. 1. Do you know where can I find that article? I wasn't able to locate it after doing a google search, I also tried looking in the database of my university's library but it doesn't show up in the search. http://mathoverflow.net/questions/42512/awfully-sophisticated-proof-for-simple-facts/42514#42514 Comment by Adrián Barquero Adrián Barquero 2010-10-18T01:48:21Z 2010-10-18T01:48:21Z @muad Are you sure something similar works for $n = 4$? I mean, the genus of the Fermat curve $x^n + y^n = z^n$ is $\frac{(n-1)(n-2)}{2}$ so when $n = 4$ the genus of the curve is 3. http://mathoverflow.net/questions/38856/jokes-in-the-sense-of-littlewood-examples/38975#38975 Comment by Adrián Barquero Adrián Barquero 2010-09-19T15:41:20Z 2010-09-19T15:41:20Z Yes, in fact Thomas Hungerford's book Algebra calls it that way in Exercise 11 in page 121. And according to it, the terminology is due to V.O. McBrien. http://mathoverflow.net/questions/34125/is-galois-theory-necessary-in-a-basic-graduate-algebra-course/34150#34150 Comment by Adrián Barquero Adrián Barquero 2010-08-01T21:31:35Z 2010-08-01T21:31:35Z @Darij. I guess it all depends on the particular interests you may have, but since the original motivation Galois had for his research was the problem of the solution of equations by radicals, I think it is only natural that at least the unsolvability of the quintic and the use of the methods of Galois Theory to study which equations may actually be solved by radicals, be presented in an introductory exposition to Galois Theory. http://mathoverflow.net/questions/32652/elliptic-curves-and-algebraic-stacks/32655#32655 Comment by Adrián Barquero Adrián Barquero 2010-07-21T04:15:38Z 2010-07-21T04:15:38Z Since nobody has mentioned it yet, I would suggest Frances Kirwan's book &quot;Complex Algebraic Curves&quot; for an introduction to algebraic curves and Riemann Surfaces. http://mathoverflow.net/questions/28788/nontrivial-theorems-with-trivial-proofs/28795#28795 Comment by Adrián Barquero Adrián Barquero 2010-06-20T14:23:53Z 2010-06-20T14:23:53Z Thanks a lot you guys. Your replies to my initial comment are great. http://mathoverflow.net/questions/16994/linear-algebra-texts/28738#28738 Comment by Adrián Barquero Adrián Barquero 2010-06-20T03:22:26Z 2010-06-20T03:22:26Z I agree with Andrew that Charles Curtis Linear Algebra book is superior to Hoffmann and Kunze. I took my first linear algebra course from Hoffmann and Kunze, and while Curtis's book has a better exposition of the theory, I do believe that the exercises in H&amp;K are way better. http://mathoverflow.net/questions/28788/nontrivial-theorems-with-trivial-proofs/28795#28795 Comment by Adrián Barquero Adrián Barquero 2010-06-20T02:45:37Z 2010-06-20T02:45:37Z I know that this is an important theorem but, actually how many results, say in elementary number theory, depend on the fact that there's an infinite number of primes?