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? 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. Thanks! 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.
