On Aug 24, 2012, at 6:19 PM, Geoffrey Irving <irv...@naml.us> wrote: > Hello, > > I'm implementing a code generator for simulation of simplicity (a way > to handle degeneracies in exact geometric computation). This involves > constructing and analyzing polynomials over a number of coordinate > variables plus one special infinitesimal variable 'e'. For example, > here's the polynomial for a 2x2 determinant (e.g., for checking > triangle areas). We add a different power of e to each coordinate > variable in an attempt to make sure our expressions are never exactly > zero: > > sage: R.<a,b,c,d,e> = > PolynomialRing(PolynomialRing(QQ,'a,b,c,d'),'e',sparse=True) > sage: p = (a+e**2**(0+1))*(d+e**2**(10+2))-(b+e**2**(10+1))*(c+e**2**(0+2)); p > e^4098 + a*e^4096 - e^2052 - c*e^2048 - b*e^4 + d*e^2 - b*c + a*d > > In order to avoid all degeneracies, this function must be nonzero in > the limit of small but nonzero e regardless of the values of the other > variables. To check this, I need to know whether the system of > polynomial equations defined by the coefficients of the distinct > powers of e is solvable over the other variables. What is the easiest > way to do that? Specifically? > > 1. How do I extract the coefficients of e as an ordered list? I > thought p.coefficients() would do it since I constructed the > polynomial ring nested, but p.coefficients() treats the ring as > flattened. Is that an artifact of the special assignment notation I > used to generate the ring?
Yep, it works fine if I avoid the .< stuff. > 2. Once I get this list of polynomials, how I check whether it's > solvable over the reals? Ideally it will be unsolvable, but for > debugging purposes I want to know some of the solutions if any do > exist. Actually, it may turn out that for all practical cases one of the coefficients of e is a nonzero constant, in which case a solve is entirely unnecessary. I would still like to know how to handle the all nonconstant case, though. Geoffrey -- You received this message because you are subscribed to the Google Groups "sage-support" group. To post to this group, send email to sage-support@googlegroups.com. To unsubscribe from this group, send email to sage-support+unsubscr...@googlegroups.com. Visit this group at http://groups.google.com/group/sage-support?hl=en.
