Timeline for (Row?) reduction of a system of equations
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 17, 2019 at 18:38 | comment | added | Daniel Lichtblau | Newer reference. | |
| Sep 17, 2019 at 18:37 | vote | accept | leastaction | ||
| Sep 17, 2019 at 18:35 | comment | added | Daniel Lichtblau | (2) As for reference in the context of this problem, I do not know. I kinda rolled my own on this one. A reference for using this basic approach in the context of integer linear programming would be chapter 2 section 8 of the book "An Introduction to Gro"bner Bases" by Adams and Loustaunau. They in turn base it on a 1991 article by Conti and Traverso. There is also this Mathematica notebook by my colleague Devendra Kapadia in our cyber-library. | |
| Sep 17, 2019 at 18:31 | comment | added | Daniel Lichtblau | (1) I should restate matters. I do not know that this will fail in general cases. I'm just not certain it will always give the desired result, if, say, there are other equations in the ideal of total degree 2. But maybe that is ruled out for some reason. | |
| Sep 17, 2019 at 17:23 | comment | added | leastaction | I think your point is that after the last step of transposing the $r_0$ and $e_i$ terms in the output of your code to the right-hand side, I would get the desired form. Is there a reference for this procedure? I'll be happy to discuss it more particularly because these equations originate as moment map conditions of the resolution of a toric singularity. | |
| Sep 17, 2019 at 17:20 | comment | added | leastaction | Thanks for your detailed reply, Daniel. The objective of the <black box> is to take those $(N+1)$ linear equations and express a certain number $K$ of them into the form $p_i - 2 p_j + p_k = f(r_0, \xi_1, \xi_2, \ldots)$ and the rest $(N-K+1)$ into the form $p_\alpha - p_\beta = g(r_0, \xi_1, \xi_2, \ldots)$. If you wish, the $(N+1)^{th}$ input equation is a "pivot". Indeed these equations arise from a toric geometry even in the my application. And the number $K$ is characteristic of the underlying geometry. Why do you think this approach is not expected to work in general? | |
| Sep 17, 2019 at 15:21 | history | answered | Daniel Lichtblau | CC BY-SA 4.0 |