Thank you for the responses.
I developed this equation as part of an integer factorization toolkit. Complex and rational values apply to `a,b,c,d,x,y` which can also resolve this equation to an integer value.
I can solve for the `a,b,c,d` values in a deterministic manner but resolving `x,y` for large values of `F` in polynomial time still evades me. I am somewhat stymied and vexed regarding how to solve this equation without using sieving or random processes for large values. Transformations into polar coordinates, complex analysis and differential geometry are valid solution paths. A suitable selection for the values `a,b,c,d,x` will algebraically factor the equivalence into (m*y+c1)*(n*y+c2). ie. a=0:b=1:c=35:d=40:x=1 >(17*y+15)*(2130*y+97)
F=2*x^3+(19/260*a60*b*y)*x^2+(21/2+314*a+30*d*y+30*c+314*b*y)*x+2+205*a+11*c+900*b*y*c+420*a*b*y+210*a^2+210*b^2*y^2+11*d*y+900*a*d*y+900*b*y^2*d+205*b*y+900*a*c
where F can be any integer.
Basically, my question is, what are the least number of variables within this equation that must be known and what are their numeric limits before it cannot be solved in polynomial time. Conversely, what must be known before this equation can be solved in polynomial time and what tools are required. ie LLL, infinite precision, etc...
Last fiddled with by jwaltos on 20170914 at 12:47
Reason: clarification
