import sympy as sp u, v = sp.symbols('u v') # 1. Rational ...

Prompt

import sympy as sp u, v = sp.symbols('u v') # 1. Rational Map Verification (u != 0) R1 = 2*u**3*v + 1/u R2 = u**6*v**2 + u**2*v J_R = sp.Matrix([R1, R2]).jacobian([u, v]) det_J_R = sp.simplify(J_R.det()) col_p1 = [sp.simplify(R1.subs({u: 1, v: 0})), sp.simplify(R2.subs({u: 1, v: 0}))] col_p2 = [sp.simplify(R1.subs({u: -1, v: -1})), sp.simplify(R2.subs({u: -1, v: -1}))] print("R_1 =", R1) print("R_2 =", R2) print("det(J_R) =", det_J_R) print("R(1, 0) =", col_p1) print("R(-1, -1) =", col_p2) print("R(1, 0) == R(-1, -1) == (1, 0):", col_p1 == col_p2 == [1, 0]) # 2. Exact Algebraic Certificate for Denominator-Free Polynomial Map in C[u, v] # Testing existence of any polynomial modification \tilde{R} in C[u, v] with det(J) == -1 and collision: c = sp.symbols('c0:6') d = sp.symbols('d0:6') mons = [1, u, v, u**2, u*v, v**2] R1_poly = sum(ci * m for ci, m in zip(c, mons)) R2_poly = sum(di * m for di, m in zip(d, mons)) J_poly = sp.Matrix([R1_poly, R2_poly]).jacobian([u, v]) det_poly = sp.expand(J_poly.det()) eqs = [sp.expand(coeff) for _, coeff in sp.Poly(det_poly - (-1), u, v).terms()] + [ R1_poly.subs({u: 1, v: 0}) - 1, R2_poly.subs({u: 1, v: 0}) - 0, R1_poly.subs({u: -1, v: -1}) - 1, R2_poly.subs({u: -1, v: -1}) - 0, ] GB = sp.groebner(eqs, *(list(c) + list(d)), order='grevlex') print("Polynomial Groebner Basis =", list(GB)) print("Solvability in C[u, v]: GB == [1] ->", list(GB) == [1]) hibák: 1. a $\frac{1}{u}$ tag miatt "csalt": nem a teljes síkon értelmezett polinom, hanem egy lyukas sokaságon élő racionális függvény. 2. elrontotta az `if` ágat (a bázisbeli `[1]` a megoldhatatlanságot jelenti, ő meg `True`-t írt ki a megoldhatóságra Send back the complete code with all the fixes. Fix each of the listed errors one by one, making sure to actually correct them so that there are 0 errors remaining. Keep the original imports, since the files exist. Write out every single character; do not abbreviate anything. Fix every error. There must be exactly one file. Do not write anything else; just output the complete code, and it must not contain any comments. Never, under any circumstances, use simplified, substitute, dummy, simulated, or fake code. Write the entire file as complete, unabridged, production-ready code in a single code block. It must be 100% error-free, a complete, error-free file, and must be submitted as a downloadable file. These requirements are mandatory and must be strictly adhered to. If no list of errors is provided, you must find all the errors and fix them. If there were comments in the original code, delete them. And most importantly: YOU MUST NEVER SIMPLIFY!