# Weighted lifts from the Jacobian counterexample Date: 2026-07-20 This note records deductions made while building the visual guide. They are exactly checked in `verify.py`, but no claim of literature priority is made. ## 1. The announced map as a weighted lift Put ```text v = xy, t = x^2 z. ``` The announced map has the form ```text F = (alpha(v,t)/x^2, beta(v,t)/x, x gamma(v,t)), ``` where ```text alpha = (1+v)^3 t + v^2(1+v)(4+3v), beta = v + 3(1+v)^2 t + 3v^2(4+3v), gamma = 2-3v-t. ``` The coordinate change `(x,y,z) -> (x,v,t)` has Jacobian `x^3`. Consequently the three-dimensional Jacobian is the two-variable determinant ```text det [ -2 alpha alpha_v alpha_t ] [ -beta beta_v beta_t ] [ gamma gamma_v gamma_t ] = -2. ``` If ```text P = beta gamma, Q = alpha gamma^2, ``` then, for every weighted lift of this form, ```text Jac(P,Q) = -det(JF) gamma^2. ``` For the announced map, `u=1+v` satisfies ```text (P^3-P^2-18PQ+27Q^2+16Q)u^3 + (P^2-12Q)u - 4Q = 0. ``` This gives generic fiber degree three. The exact target `(-8/27,0,1)` has three preimages, listed and checked in `verify.py`. ## 2. Tangent geometry and the branch at infinity After harmless constant rescalings of the output invariants, the announced map is encoded by ```text p(w) = 2w - 3w^2, q(w) = w^2 - 2w^3. ``` These satisfy ```text q'(w) = w p'(w). ``` Consequently the plane curve ```text Gamma(w) = (p(w),q(w)) ``` has tangent slope `dq/dp=w`. Its tangent at parameter `w` passes through an output point `(P,Q)` precisely when ```text Q - q(w) = w(P - p(w)). ``` Since ```text R(w) = w p(w) - q(w) = integral_0^w p(s) ds, ``` this is exactly the hidden inverse equation ```text R(w) = wP - Q. ``` Thus inverse images correspond to tangents from `(P,Q)` to `Gamma`. A repeated inverse root must satisfy both ```text R(w)-wP+Q = 0, p(w)-P = 0. ``` It follows that `(P,Q)=(p(w),q(w))`. In the weighted lift this is exactly the locus `gamma=0`, while reconstruction uses ```text x = C/gamma. ``` For `C != 0`, the would-be collision of inverse roots therefore occurs only when the corresponding input escapes to infinity. This is the geometric mechanism by which the three-dimensional map remains etale at every finite point. For fixed nonzero `C`, this gives an explicit family of asymptotic values ```text (A,B,C) = (q(w)/C^2, p(w)/C, C) ``` up to the constant output normalizations used above. Determining the full closure, including all `C=0` boundary components, is a next problem; equality with the complete nonproper-value set has not yet been claimed. ## 3. A general one-variable construction Fix a nonzero constant `c`. Choose a polynomial `p(w)` satisfying ```text p(0) = 0, p(1) = -c, integral_0^1 p(w) dw = 0. ``` Define `q(0)=0` and ```text q'(w) = w p'(w) / c. ``` Let `kappa=p'(1)/c`, assume `kappa != -2`, and set ```text a = -(1+kappa)/(2+kappa). ``` For any nonzero `b`, put ```text v = xy, t = x^2 z, u = 1+v, gamma = 1 + a v + b t, w = u gamma, beta = c + p(w)/gamma, alpha = u + q(w)/gamma^2. ``` Because `p(0)=0`, the expression `p(w)/gamma` is polynomial in `(u,gamma)`. Because `q` has a double zero at zero, `q(w)/gamma^2` is also polynomial. The endpoint and integral conditions imply ```text beta(0,0) = 0, alpha(0,0) = 0, partial alpha / partial v (0,0) = 0. ``` It follows that ```text F_p(x,y,z) = (alpha(v,t)/x^2, beta(v,t)/x, x gamma(v,t)) ``` is a polynomial map. To calculate its Jacobian, use quotient coordinates ```text P = beta gamma = c gamma + p(w), Q = alpha gamma^2 = w gamma + q(w). ``` In coordinates `(w,gamma)`, ```text Jac(P,Q) = -c gamma. ``` The changes `(v,t)->(u,gamma)->(w,gamma)` have determinants `b` and `gamma`. Hence ```text Jac(P,Q) with respect to (v,t) = -bc gamma^2, det(J F_p) = bc. ``` Finally, eliminating `gamma` gives ```text integral_0^w p(s) ds = wP - cQ. ``` If `p` has degree `d`, the generic fiber degree is `d+1`. ## 4. An explicit generic-degree-four map For a particularly small collision, take the raw one-variable data ```text c = 1, p(w) = w - 2w^3, q(w) = (1/2)w^2 - (3/2)w^4, a = -4/3, b = -1. ``` The raw lift has `u=1+xy`, `gamma=1-(4/3)xy-x^2z`, determinant `-1`, and rational coefficients. Precompose by the linear change `y -> 3y` and scale the first output by `2`. This gives the integer-coefficient map ```text u = 1+3xy, gamma = 1-4xy-x^2z, A = (2u+u^2-3u^4 gamma^2)/x^2, B = (1+u-2u^3 gamma^2)/x, C = x gamma. ``` The apparent quotients cancel. The resulting polynomial map has: ```text component degrees: (12,11,4), Jacobian determinant: -6, generic fiber degree: 4. ``` It sends both ```text (1,0,0), (-1,0,2) ``` to `(0,0,1)`. Generic fiber degree distinguishes this map from the announced degree-three example under left and right polynomial coordinate changes. For a complete generic-degree certificate, put ```text P = B C, Q = A C^2, w = u gamma. ``` Then ```text w^4 - w^2 + 2Pw - Q = 0. ``` Conversely, on the generic open set a root reconstructs the unique input by ```text gamma = P-w+2w^3, u = w/gamma, x = C/gamma, y = (u-1)/(3x), z = (1-(4/3)(u-1)-gamma)/x^2. ``` At the target `(A,B,C)=(1,0,1)`, the quartic is ```text w^4-w^2-1 = 0. ``` Its discriminant is `-400`; its four distinct roots reconstruct four distinct inputs, all checked exactly in `verify.py`. ## 5. Every generic fiber degree at least three For every integer `n >= 3`, put `d=n-1` and take the single explicit seed ```text p_d(w) = 2w - 3w^2 + w(1-w)(w^(d-2) - 6/(d(d+1))). ``` For `d=2`, the last term vanishes identically and this is the quadratic seed of section 2. For `d>=3`, its leading coefficient is `-1`, so it has degree `d`. In every case, ```text p_d(0)=0, p_d(1)=-1, integral_0^1 p_d(w) dw=0, p_d'(1)=-5+6/(d(d+1)) != -2. ``` Apply the construction of section 3 with `b=c=1`. It gives an explicit polynomial map ```text F_n : C^3 -> C^3, det JF_n = 1. ``` Its inverse equation has degree `d+1=n`: ```text integral_0^w p_d(s) ds = wP-Q. ``` Away from its discriminant, this equation has `n` distinct roots. A repeated root is exactly where `gamma=P-p_d(w)=0`; thus on the further open set `C!=0`, each simple root has `gamma!=0` and reconstructs one distinct input. Hence ```text [C(x,y,z) : C(F_{n,1},F_{n,2},F_{n,3})] = n, ``` or equivalently a Zariski-generic target has exactly `n` complex preimages. This proves that every generic fiber degree `n>=3` occurs in the fixed dimension three. The proof is uniform in `n`; the finite checks in the scripts are regression tests, not the basis for the all-`n` conclusion. ## 6. Rational certificates for degrees 3 through 100 The all-`n` seed above is concise, but its generic inverse roots need not be rational. For a deliberately redundant finite certificate, one can instead choose the inverse polynomial first and make all of its roots integers. For `3 <= n <= 100`, define a set of `n-2` roots by ```text S_n = {2} union {j,1-j : 3 <= j <= (n+1)/2} (n odd), S_n = {3,4} union {j,1-j : 5 <= j <= (n+4)/2} (n even), ``` where an upper limit below the lower limit means that the second set is empty. Put ```text G_n(w) = w(w+1) product_(r in S_n) (w-r), lambda_n = G_n'(0)-G_n'(1), H_n(w) = G_n(w)/lambda_n. ``` Every paired pair `{j,1-j}` contributes one to the ratio `S_n(0)/S_n(1)`. The unpaired root `2`, or the pair `3,4`, contributes two. Thus `S_n(0)=2S_n(1)`, and consequently ```text G_n(1)=G_n'(0). ``` The exact atlas checker verifies, for each `3 <= n <= 100`, that `lambda_n != 0` and `H_n''(1) != -2`. Define ```text p_n(w) = H_n'(w)-H_n'(0), q_n(w) = w H_n'(w)-H_n(w), a_n = -(1+H_n''(1))/(2+H_n''(1)). ``` Then ```text p_n(0)=0, p_n(1)=-1, integral_0^1 p_n(w)dw=0. ``` Applying section 3 with `b=c=1` gives a polynomial map `F_n` with `det JF_n=1` and generic fiber degree `n`. These maps have an especially simple exact fiber. Set ```text T_n = (0,-H_n'(0),1). ``` For each of the `n` distinct integer roots `r` of `G_n`, put ```text gamma_r = -H_n'(r), X_(n,r) = ( 1/gamma_r, r-gamma_r, (gamma_r-1-a_n(r/gamma_r-1)) gamma_r^2 ). ``` All coordinates are rational. Since the roots are simple, `gamma_r != 0`. Exact substitution gives ```text F_n(X_(n,r)) = T_n. ``` Conversely, over `T_n` the inverse equation is exactly `H_n(w)=0`, and the reconstruction above is unique for every root. Therefore the displayed fiber has exactly `n` points, not merely at least `n`. Because the inverse equation has degree `n` and this fiber consists of `n` simple points, the generic fiber degree is also `n`. At `n=3`, the roots are `0,-1,2` and the construction gives `p_3(w)=2w-3w^2`, the minimal seed underlying the announced map. The atlas uses the determinant-one normalization `b=1`; Alpöge's displayed map uses a different nonzero `b` and a linear output scaling. The standard-library script `verify_counterexamples.py` checks all 98 maps and all 5,047 rational preimages using exact `Fraction` arithmetic. This finite atlas is an audit aid; section 5 remains the proof for every `n>=3`, including values beyond 100.