read the full code. list every of any type error! just list ...

Prompt

read the full code. list every of any type error! just list the results, nothing else. import sympy as s x,y=s.symbols('x y') a,b,c,d,e,f,g,h=s.symbols('a b c d e f g h') P=a*x**3+b*x**2*y+c*x*y**2+d*y**3 Q=e*x**3+f*x**2*y+g*x*y**2+h*y**3 J=s.Poly(s.expand(s.diff(x+P,x)*s.diff(y+Q,y)-s.diff(x+P,y)*s.diff(y+Q,x)-1),x,y) eq=[s.factor(co) for _,co in J.terms()] print('Jacobian coefficient equations:',*eq,sep='\n') G=s.groebner(eq,a,b,c,d,e,f,g,h,order='grlex') print('Groebner basis size:',len(G.polys)) print('Groebner basis:') for p in G.polys: print(' ',s.factor(p.as_expr())) # For every residual coefficient in F(F^{-1}) and F^{-1}(F), reduce its parameter coefficient modulo the exact constraint ideal. Xm=x-P; Ym=y-Q # F o (Id-H), residual of H(x-H)-H(x) Pminus=s.expand(P.subs({x:Xm,y:Ym}, simultaneous=True)-P) Qminus=s.expand(Q.subs({x:Xm,y:Ym}, simultaneous=True)-Q) # (Id-H) o F residual, also H(x+H)-H(x) Pplus=s.expand(P.subs({x:x+P,y:y+Q}, simultaneous=True)-P) Qplus=s.expand(Q.subs({x:x+P,y:y+Q}, simultaneous=True)-Q) def check(poly): poly=s.Poly(poly,x,y) rems=[] for coeff in poly.coeffs(): rem=G.reduce(s.expand(coeff))[1] rems.append(s.expand(rem)) return all(r==0 for r in rems),rems for label,polys in [('F∘(Id−H)',(Pminus,Qminus)),('(Id−H)∘F',(Pplus,Qplus))]: results=[check(z) for z in polys] print(label, 'both components vanish modulo constraints:', all(ok for ok,_ in results)) print(' remainders:', [len(set(str(r) for r in rr)) for _,rr in results], 'nonzero coefficient remainders:',sum(r!=0 for _,rr in results for r in rr))