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2018-03-19 19:15:21 +0100	received badge	● Famous Question (source)
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2016-04-03 02:06:35 +0100	commented answer	How to implement the multivariable division algorithm without passing to Grobner bases? Also, for what it's worth, this is the problem I'm tackling: Take the 3-adic valuation on Q and extend it to a discrete valuation v on Q(x) by assigning the value v(x)=1/2, and then extend that by setting v(x^2+1)=3, where we define the value of a polynomial f(x) by using the decomposition f(x)=\sum a_i(x)(x^2+1)^i where each a_i(x) is of degree less than 2, which is unique and therefore makes the valuation work. Do the same in the variable y and the polynomial y^3+y+1 to get a discrete valuation w. My research question: what discrete valuations are there that extend both of these valuations v and w to a discrete valuation on Q(x,y)? Is there a canonical choice? To do this, I'm needing to break up ... (more)
2016-04-03 01:57:47 +0100	commented answer	How to implement the multivariable division algorithm without passing to Grobner bases? Thanks, Bruno! A question about the things you've defined: you're defining these things for L an arbitrary list, right? How does one set this code up so that your definitions of list_quo_rem and remainder recognize that L is an arbitrary list?
2016-04-02 20:37:58 +0100	received badge	● Student (source)
2016-04-01 21:27:49 +0100	commented answer	How to implement the multivariable division algorithm without passing to Grobner bases? As a corollary question: would you happen to know how I could get all possible remainders from all possible orders with one command? If I'm dividing 5 or 6 things at once, iterating the program you've written to get all possible remainders isn't very practical to type out. It looks like I might need to do that to test the examples I've come up with to test my conjecture. Thanks!
2016-04-01 21:03:24 +0100	commented answer	How to implement the multivariable division algorithm without passing to Grobner bases? Great! That makes sense. Thanks again!
2016-04-01 18:00:31 +0100	commented answer	How to implement the multivariable division algorithm without passing to Grobner bases? Thank you so much! I wish I could at least award you karma for this, but it seems I can't since I don't have enough myself. One more question: How are we defining the object L? Is it the ideal generated by the pair q and r, is it an ordered list, or is it something else? Thanks again!
2016-04-01 17:09:26 +0100	received badge	● Scholar (source)
2016-03-31 22:37:04 +0100	asked a question	How to implement the multivariable division algorithm without passing to Grobner bases? Hi, I'm new to Sage, and I'm wondering how to implement the multivariable division algorithm in Sage. I pulled up the "Multivariate Polynomials via libSINGULAR" page of the Sage Reference Manual v7.1, but it wasn't helpful. What I'm wanting is a generalization of the quo_rem command that can take in more than one argument on the right and follows the division algorithm with respect to a fixed monomial ordering and the order that the polynomials are entered in. Is there any set of commands that does that for me? If so, would you please include the code, say for the following example? Divide the polynomial yx^2 + xy^2 + y^2 by xy-1 and y2 -1 (in that order) using the lexicographic ordering with x>y. I would like to process more complicated examples, perhaps with that order and dividing by 8 things at once rather than 2. I've learned about the p.mod(I) and p.reduce(I) commands where p is a polynomial and I is an ideal. The problem with those is that they seem to pass to a Grobner basis for I to get a "canonical" remainder rather than the remainder we'd get from the given order of the polynomials, as I tested switching the order of the polynomials in defining an ideal I and it did not change my answer for p.mod(I) or p.reduce(I). Thanks!