I have been trying to simplify this expression but with no luck. I checked using PARI, it indeed gives 0, means the change of variable p and v do satisfy the curves given. Can someone drop me a hint on how to work this out.
> p,x,v,y= var('p x v y')
> p=(-72*x-264+36*y)/(9*x^2+30*x-119)
> v=(-162*x^4+540*x^3-648*x^2*y+13176*x^2-4752*x*y+62340*x-16488*y+153994)/(81*x^4+540*x^3-1242*x^2-7140*x+14161)
> eq1=expand(v^2-(p^4-2*p^3+5*p^2+8*p+4))
> eq=eq1.subs({y:sqrt(-x^3+(121/3)*x+(1690/27))}).subs({sqrt(-x^3+(121/3)*x+(1690/27)):y})
> eq2=eq.simplify_full();eq2
The output gave :
3456*(27*(-x^3 + 121/3*x + 1690/27)^(3/2)*(9*x^2 + 174*x + 409) + (243*x^5 + 4698*x^4 + 1242*x^3 - 204696*x^2 - 739461*x - 691210)*y)/(6561*x^8 + 87480*x^7 + 90396*x^6 - 2498040*x^5 - 3874554*x^4
+ 33029640*x^3 + 15803676*x^2 - 202219080*x + 200533921)
It is suppose to be zero (I have checked). Problem lies at the expression 3456*(27*(-x^3 + 121/3*x + 1690/27)^(3/2). This term should be simplified into somewhat y^2*y=(-x^3 + 121/3*x + 1690/27). It will then simplify and give 0. But how to work that out.
