Timeline for Solution of a special class of Diophantine Equations
Current License: CC BY-SA 3.0
22 events
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| Jun 15, 2013 at 14:31 | history | edited | XL _At_Here_There | CC BY-SA 3.0 |
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| Jun 8, 2013 at 4:53 | comment | added | XL _At_Here_There | They are not all algebraic,and in some cases they are all transcendental. | |
| May 31, 2013 at 9:54 | comment | added | XL _At_Here_There | @S. Carnahan,and let me think about it carefully later | |
| May 31, 2013 at 9:12 | comment | added | XL _At_Here_There | @S. Carnahan,encyclopediaofmath.org/index.php/Algebraic_function Have I misunderstood what parametrization of algebraic function is ?Or the function like $$f_1(x),\cdots,f_n(x)$$ are not algebraic? | |
| May 31, 2013 at 5:43 | comment | added | S. Carnahan♦ | I don't see why you should expect automorphic functions to parametrize arbitrary varieties. If you have a smooth plane curve $P(x,y)=0$ of positive genus, you cannot write $y$ as a function of $x$ without choosing branches. A standard example is $y^2 = x^3+1$, where $y=\pm \sqrt{x^3+1}$ is not really a function of $x$, and is not automorphic (as far as I know). | |
| May 31, 2013 at 4:03 | history | edited | XL _At_Here_There | CC BY-SA 3.0 |
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| May 31, 2013 at 3:58 | history | edited | XL _At_Here_There | CC BY-SA 3.0 |
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| May 31, 2013 at 3:49 | history | edited | XL _At_Here_There | CC BY-SA 3.0 |
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| May 31, 2013 at 3:42 | history | edited | XL _At_Here_There | CC BY-SA 3.0 |
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| May 30, 2013 at 22:55 | comment | added | XL _At_Here_There | @all,and such Diophantine equations may be a system of equation,and we suppose we just ask about other variables' solutions which is a function of one variable in the equations or equation,there are two variables in Diophantine equations or more,three variables is also just a special case .it is irrelevent whether such Diophantine equations have integer as solution or not. | |
| May 30, 2013 at 22:44 | comment | added | XL _At_Here_There | @quim,@François,@Charles,@ Dietrich ,we just suppose other variables can have a function of x ,and the functions satisify the Diophantine equations,I do not know what functions they are,some of them may be rational ,lik the example,some of them may be algebraic ,whether implicit or explicit,is it possible that some of them transcendental?.Dietrich,I am not asking about parametrization of solution,I know parametrized solutions may be transcendental,like modular function. | |
| May 30, 2013 at 18:12 | comment | added | Dietrich Burde | Do you mean parametrized solutions $(x(t),y(t),z(t)$ of a Diophantine equation F(x,y,z)=0$ ? | |
| May 30, 2013 at 15:00 | comment | added | Charles Matthews | So we have x, f(x), g(x), h(x) ... where f, g, h are certain functions. What now? Which equations are concerned? I have a feeling you want something about implicit algebraic functions. | |
| May 30, 2013 at 14:35 | comment | added | François Brunault | I don't understand your question. Could you clarify? | |
| May 30, 2013 at 14:29 | comment | added | quim | If each variable can be written as a rational function of x, then the curve has genus zero. If not, then what does "every other variable has function of x the variable as it's solution" mean? | |
| May 30, 2013 at 13:27 | comment | added | XL _At_Here_There | @Charles,I think there must be some misunderstanding,I think I am not asking about question of curves of genus 0.maybe some of them are about curves of genus 0.The example is just a very special case. | |
| May 30, 2013 at 13:23 | history | edited | XL _At_Here_There | CC BY-SA 3.0 |
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| May 30, 2013 at 13:05 | comment | added | XL _At_Here_There | @François,thank you very much for your edit | |
| May 30, 2013 at 12:42 | comment | added | Charles Matthews | The question really isn't well posed. The theory of diophantine equations for all curves of genus 0 (i.e. all "space curves" that are rational curves from a geometric point of view) was written down by Hilbert and Hurwitz. It is in Mordell's books. If that is not essentially the question you are asking, I'm not really understanding the question. | |
| May 30, 2013 at 11:19 | history | edited | François G. Dorais |
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| May 30, 2013 at 11:16 | history | edited | XL _At_Here_There |
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| May 30, 2013 at 11:08 | history | asked | XL _At_Here_There | CC BY-SA 3.0 |