I have two equations, two unknown (ISP, ch) and one parameter ($n_C$). My aim is to find the values of the unknown according to the parameter. The original rational equations are here : https://framabin.org/p/?5a4a682c865ca030#GHL5aWQ6hPZ6m0csuQwQ7GSj15UWJxliKIohqkj21gY=
I converted them in polynomial, you can look at them here in a $ \LaTeX$ format: https://v1.overleaf.com/23360386gqfrrchwvyxx
I used singular through sagemath to get the Gröbner basis. Nonetheless, singular crashes to get the variety of the ideal. But
The code I used in sagemath :
pch_simple = final2_simple.simplify_rational().rhs().denominator() * ch - final2_simple.simplify_rational().rhs().numerator() #polynom for ch
pisp_simple = final1_simple.simplify_rational().rhs().denominator() * ISP - final1_simple.simplify_rational().rhs().numerator() #polynom for ch
Rsimple = PolynomialRing(FractionField(PolynomialRing(QQ, names='n_C')), names='ISP,ch') #Multivariate Polynomial Ring in ISP, ch over Fraction Field of Univariate Polynomial Ring in n_C over Rational Field
plist_simple = [Rsimple(pch_simple),Rsimple(pisp_simple)] #list (system) of my polynomials
I_simple = ideal(plist_simple) #the ideal of the system of polynomials
G_simple = I_simple.groebner_basis() #the Groebner basis
len(G_simple) #I have three equations in my Groebner basis
I_simple.dimension() #0, the ideal is zero-dimensional
I_simple.variety() #stuck
I think the problem come from the high degree of my second equation ('pisp_simple'). In this equation, ISP is degree two but ch is degree 15. In the first equation ('pch_simple'), ISP is degree one and ch is degree three.
Must I concede my polynomial is impossible to solve or does alternative methods exist to solve my nonlinear system of equations composed by two polynomials ?