Timeline for Positive solutions of linear Diophantine equations
Current License: CC BY-SA 2.5
14 events
when toggle format | what | by | license | comment | |
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Jan 10, 2013 at 11:49 | answer | added | Tim Seguine | timeline score: 0 | |
May 9, 2010 at 16:21 | comment | added | SIB | Thanks to you all for the discussion, answers, and references. It is really helpful. | |
S May 9, 2010 at 16:16 | vote | accept | SIB | ||
May 10, 2010 at 11:32 | |||||
May 9, 2010 at 16:15 | vote | accept | SIB | ||
S May 9, 2010 at 16:16 | |||||
May 9, 2010 at 15:27 | answer | added | Torsten Ekedahl | timeline score: 4 | |
May 9, 2010 at 13:42 | answer | added | Alexey Ustinov | timeline score: 5 | |
May 9, 2010 at 13:25 | comment | added | Sidney Raffer | Right: Thanks to Sergei Ivanov we can forget the "sufficiently large coordinate" idea. BUT -- The relevance of Ramirez-Alfonsin is that under some circumstances, we get a whole cone of representable b's. | |
May 9, 2010 at 9:44 | comment | added | Pete L. Clark | It seems to me that the theorem discussed in Section 6.5 of Ramirez-Alfonsin's book is not directly relevant to the question at hand. For instance, the matrix in Sergei Ivanonv's answer below satisfies the determinant condition (6.7) and thus admits a "pseudo-conductor". But this does not mean that every $b$ with sufficiently large coordinates is represented by non-negative integers. | |
May 9, 2010 at 9:03 | comment | added | Wadim Zudilin | @SJR: Thanks for the tip, I'll check whether it really covers the general matrix case, not only $k=1$ as in Pete Clark's solution below. | |
May 9, 2010 at 9:03 | answer | added | Sergei Ivanov | timeline score: 11 | |
May 9, 2010 at 8:57 | comment | added | Sidney Raffer | There is a detailed discussion of your question in Section 6.5 of Alphonsin's book "The Diophantine Frobenius Problem" | |
May 9, 2010 at 8:50 | history | edited | Pete L. Clark |
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May 9, 2010 at 8:11 | comment | added | Wadim Zudilin | I don't think that the problem is specially treated (some time ago I had a related one and nobody could suggest something concrete). You can write a general solution in the form $x_0+Ct$ where $x_0$ is a fixed solution and $Ct$ ($C$ a matrix and $t$ run over $\mathbb Z^l$) is a general solution of $Ax=0$. Then the required nonnegativity poses conditions on components $x_0+Ct$ (linear programming?). Since $x_0$ depends on $b$, you might be able to verify you expectation about large entries of $b$. | |
May 9, 2010 at 5:37 | history | asked | SIB | CC BY-SA 2.5 |