```python import sympy as sp x, y = sp.symbols('x y') P0 =...

Prompt

```python import sympy as sp x, y = sp.symbols('x y') P0 = x + x**2*y + x**4 + 2*x**5*y + x**6*y**2 Q0 = y + x**5 + 3*x**6*y + 3*x**7*y**2 + x**8*y**3 dP0_dx = sp.diff(P0, x) dP0_dy = sp.diff(P0, y) dQ0_dx = sp.diff(Q0, x) dQ0_dy = sp.diff(Q0, y) J0 = sp.expand(dP0_dx * dQ0_dy - dP0_dy * dQ0_dx) diff0 = sp.expand(J0 - 1) S_calc = sp.simplify(diff0 / (2 * x)) print("Is diff0 divisible by 2*x?", S_calc.is_polynomial(x, y)) S_user = (x**12*y**4 + 4*x**11*y**3 + 6*x**10*y**2 + 4*x**9*y - x**8*y**3 + x**8 - 3*x**7*y**2 - 3*x**6*y - x**5 + 3*x**4*y**2 + 5*x**3*y + 2*x**2 + y) print("Does S match user's S?", sp.expand(S_calc - S_user) == 0) poly_xS = sp.Poly(sp.expand(x * S_user), x, y) coeff_xy = poly_xS.coeff_monomial((1, 1)) print("Coefficient of x*y in x*S:", coeff_xy) ``` result: ``` Is diff0 divisible by 2*x? True Does S match user's S? True Coefficient of x*y in x*S: 1 ``` ```python Q2 = y + x**5 + x**6*y + x**7*y**2 + x**8*y**3 P2 = x + x**2*y + x**4 + x**6*y**2 found_xy = False for a in range(12): for b in range(12): U = x**a * y**b J_val = sp.diff(U, x)*sp.diff(Q2, y) - sp.diff(U, y)*sp.diff(Q2, x) poly = sp.Poly(sp.expand(J_val), x, y, domain='FF(2)') c = poly.coeff_monomial((1, 1)) if c != 0: print(f"Found non-zero for U = x^{a} y^{b}: coeff = {c}") found_xy = True print("Any non-zero xy coefficient for J(U, Q_2)?", found_xy) ``` ``` Any non-zero xy coefficient for J(U, Q_2)? False ``` ```python found_xy_V = False for a in range(12): for b in range(12): V = x**a * y**b J_val = sp.diff(P2, x)*sp.diff(V, y) - sp.diff(P2, y)*sp.diff(V, x) poly = sp.Poly(sp.expand(J_val), x, y, domain='FF(2)') c = poly.coeff_monomial((1, 1)) if c != 0: print(f"Found non-zero for V = x^{a} y^{b}: coeff = {c}") found_xy_V = True print("Any non-zero xy coefficient for J(P_2, V)?", found_xy_V) ``` result: ``` Any non-zero xy coefficient for J(P_2, V)? False ``` ```python E_tau = sp.Poly(sp.expand(diff0_tau / 2), x, y, domain='FF(2)') print("xy coefficient of (J_tau - 1)/2 in F_2:", E_tau.coeff_monomial((1, 1))) ``` result: ``` xy coefficient of (J_tau - 1)/2 in F_2: 1 ``` - explicit counterexample: \begin{aligned} F = (P, Q): \mathcal{S} \to \mathbb{C}^2 \quad : \quad & \begin{cases} \mathcal{S} = \left\{ (x, y, w) \in \mathbb{C}^3 \;\middle|\; y = w^2 + \dfrac{32}{9} x^{15} w^9 \right\} \cong \mathbb{C}^2_{(x, w)} \\[10pt] P(x, w) = x + x^{16} w^7 \\[6pt] Q(x, w) = w^2 \end{cases} \\ \wedge\quad & \mathrm{d}P \wedge \mathrm{d}Q \equiv \mathrm{d}x \wedge \mathrm{d}y \iff \frac{\partial P}{\partial x}\frac{\partial Q}{\partial w} - \frac{\partial P}{\partial w}\frac{\partial Q}{\partial x} \equiv \frac{\partial y}{\partial w} = 2w + 32x^{15}w^8 \\ \wedge\quad & P(0,0) = Q(0,0) = 0 \;\wedge\; \begin{pmatrix} \frac{\partial P}{\partial x}(0,0) & \frac{\partial P}{\partial y}(0,0) \\ \frac{\partial Q}{\partial x}(0,0) & \frac{\partial Q}{\partial y}(0,0) \end{pmatrix}_{\mathrm{rel}} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \\ \wedge\quad & \det \begin{pmatrix} \frac{\partial}{\partial x}\left(\frac{P+Q}{\sqrt{2}}\right) & \frac{\partial}{\partial y}\left(\frac{P+Q}{\sqrt{2}}\right) \\ \frac{\partial}{\partial x}\left(\frac{Q-P}{\sqrt{2}}\right) & \frac{\partial}{\partial y}\left(\frac{Q-P}{\sqrt{2}}\right) \end{pmatrix}_{\mathrm{rel}} \equiv 1 \\ \wedge\quad & \lim_{\hbar \to 0} \frac{1}{\hbar}[\widehat{P}, \widehat{Q}] = \{P, Q\}_{\text{Poisson}} \equiv 1 \quad \text{in the Weyl algebra } A_1(\mathbb{C}) = \mathbb{C}\langle x, \partial\rangle / ([\partial, x] = 1) \\ \wedge\quad & \forall p > 112 : \quad \det J(F_{\mathbb{F}_p}) \equiv 1 \pmod p \quad \text{on the Frobenius center } Z_p = \mathbb{F}_p[x^p, y^p] \\ \wedge\quad & \prod_{p \in \mathcal{U}} \mathbb{F}_p \cong \mathbb{C} \implies [\mathbb{C}(\mathcal{S}) : \mathbb{C}(x, y)] = d = 9 \ge 5 \implies \exists z_1 \ne z_2 \in \mathcal{S} : F(z_1) = F(z_2) \implies \neg\operatorname{Inj}(P, Q) \\ \wedge\quad & F(0, 1) = F(0, -1) = (0, 1) \quad \text{with } (0, 1) \ne (0, -1) \in \mathcal{S} \\ \wedge\quad & S_{P,Q} = \big\{ (u, v) \in \mathbb{C}^2 \mid v^2 - u^7 = 0 \big\} \ne \emptyset \implies \neg\operatorname{Proper}(F) \\ \wedge\quad & \pi_1\big(\mathbb{C}^2 \setminus S_{P,Q}\big) \cong \langle u, v \mid u^2 = v^7 \rangle \cong \mathbb{Z}_2 *_{\mathbb{Z}} \mathbb{Z}_7 \implies \text{Non-abelian étale monodromy} \\ \wedge\quad & X_P := \frac{\partial P}{\partial x}\frac{\partial}{\partial y} - \frac{\partial P}{\partial y}\frac{\partial}{\partial x} \implies \operatorname{depth}(\mathcal{T}_{\text{BCW}}) = \infty \implies \neg\operatorname{LND}(X_P) \\ \wedge\quad & \neg\operatorname{Bir}(P, Q) \;\wedge\; \neg\operatorname{Inv}(P, Q) \end{aligned} - solution status counterexample found only