# Power-Weighted Lifts: Explicit Higher-Weight Noninjective Keller Maps in Three Variables

**Research draft — 21 July 2026**

## Abstract

For every pair of integers \(1\le k<d\), we construct an explicit polynomial map \(F_{k,d}:\mathbb A^3\to\mathbb A^3\) over any characteristic-zero field. Its Jacobian determinant is the constant \(-k/(k+1)\), its source torus weights are \((1,-k,-k-1)\), and its geometric generic degree is \(k(d+1)\). When \(k\) and \(d\) are odd, the maps have the uniform rational collision

\[
F_{k,d}(-1,0,2)=F_{k,d}(1,0,0)=(0,0,1),
\]

so they form an infinite explicit family of noninjective Keller maps. The specialization \((k,d)=(2,3)\) gives an expanded degree-eight example with a separately certified rational collision. Exact verification scripts are supplied for SymPy, SageMath, and Maple.

## 1. Construction

Fix integers \(1\le k<d\). Put

\[
v=x^k y,\qquad t=x^{k+1}z,\qquad u=1+v,
\]

\[
\gamma=1-\frac{d+k}{d}v-t,\qquad w=u\gamma.
\]

Define

\[
q(w)=\frac{k+1}{d-k}w^k-\frac{d+1}{d-k}w^d,
\]

\[
Q(w)=\int_0^wq(s)\,ds=\frac{w^{k+1}-w^{d+1}}{d-k},
\]

and

\[
p(w)=\frac{wq(w)-Q(w)}{k+1}.
\]

Finally set

\[
\alpha=\frac{p(w)}{\gamma^{k+1}}+\frac{u}{k+1},
\qquad
\beta=rac{q(w)}{\gamma^k}+1,
\]

and

\[
F_{k,d}(x,y,z)=
\left(
\frac{\alpha}{x^{k+1}},
\frac{\beta}{x^k},
x\gamma
\right).
\]

## 2. Polynomiality

Because \(q(w)\) is divisible by \(w^k\) and \(p(w)\) by \(w^{k+1}\), both \(q(u\gamma)/\gamma^k\) and \(p(u\gamma)/\gamma^{k+1}\) are polynomials in \((u,\gamma)\). It remains to prove divisibility by the displayed powers of \(x\).

Assign weighted orders

\[
\operatorname{wt}(v)=k,\qquad \operatorname{wt}(t)=k+1.
\]

Since \(v=x^k y\) and \(t=x^{k+1}z\), a polynomial in \((v,t)\) is divisible by \(x^m\) after substitution whenever every monomial has weighted order at least \(m\).

For \(\beta\), only the constant term has weight below \(k\). It vanishes because

\[
q(1)=-1.
\]

For \(\alpha\), possible weights below \(k+1\) are \(0\) and \(k\). The constant term vanishes because

\[
Q(1)=0,
\qquad
p(1)=-\frac1{k+1}.
\]

The weight-\(k\) coefficient is the coefficient of \(v\). Using

\[
q'(1)=-(d+k+1),
\qquad
\gamma=1-\frac{d+k}{d}v-t,
\]

a direct first-order expansion gives zero. Hence \(x^{k+1}\mid\alpha\) and \(x^k\mid\beta\), proving that all coordinates of \(F_{k,d}\) are polynomials.

## 3. Constant Jacobian theorem

### Theorem 3.1

For all integers \(1\le k<d\),

\[
\det JF_{k,d}=-\frac{k}{k+1}.
\]

### Proof

Write the target coordinates as \((A,B,C)\). Introduce target invariants

\[
P=AC^{k+1},\qquad R=BC^k.
\]

Since \(C=x\gamma\),

\[
P=\alpha\gamma^{k+1}=p(w)+\frac{w\gamma^k}{k+1},
\]

\[
R=\beta\gamma^k=q(w)+\gamma^k.
\]

From the definition of \(p\),

\[
p'(w)=\frac{wq'(w)}{k+1}.
\]

Therefore

\[
\det\frac{\partial(P,R)}{\partial(w,\gamma)}
=
\frac{k}{k+1}\gamma^{2k-1}.
\]

The coordinate changes \((v,t)\mapsto(u,\gamma)\mapsto(w,\gamma)\) have determinants \(-1\) and \(\gamma\), respectively. Hence

\[
\det\frac{\partial(P,R)}{\partial(v,t)}
=-\frac{k}{k+1}\gamma^{2k}.
\]

Also

\[
\det\frac{\partial(x,v,t)}{\partial(x,y,z)}=x^{2k+1},
\]

while

\[
\det\frac{\partial(C,P,R)}{\partial(C,A,B)}=C^{2k+1}=(x\gamma)^{2k+1}.
\]

Combining these determinants and preserving the coordinate ordering yields

\[
\det JF_{k,d}
=
\gamma^{-2k}
\det\frac{\partial(P,R)}{\partial(v,t)}
=-\frac{k}{k+1}.
\]

\(\square\)

## 4. Geometric generic degree

### Theorem 4.1

Over an algebraic closure of the ground field, the geometric generic degree of \(F_{k,d}\) is

\[
k(d+1).
\]

### Proof

Eliminating \(\gamma^k=R-q(w)\) from the two invariant equations gives

\[
P=\frac{wR-Q(w)}{k+1},
\]

or

\[
Q(w)-Rw+(k+1)P=0.
\]

Because \(Q\) has degree \(d+1\), this equation has degree \(d+1\) in \(w\). On the nonempty Zariski-open locus where it is separable and where \(R-q(w)\ne0\) at every root, it has \(d+1\) distinct roots and each root gives exactly \(k\) distinct solutions of

\[
\gamma^k=R-q(w).
\]

For \(C\ne0\), each pair \((w,\gamma)\) reconstructs one source point:

\[
u=\frac{w}{\gamma},\quad
x=\frac{C}{\gamma},\quad
v=u-1,
\]

\[
t=1-\frac{d+k}{d}v-\gamma,
\quad
y=\frac{v}{x^k},
\quad z=\frac{t}{x^{k+1}}.
\]

Thus the geometric generic fiber has exactly \(k(d+1)\) points. \(\square\)

## 5. Infinite explicit noninjective subfamily

### Theorem 5.1

If \(k\) and \(d\) are odd integers with \(1\le k<d\), then

\[
F_{k,d}(-1,0,2)=F_{k,d}(1,0,0)=(0,0,1).
\]

### Proof

At both points, \(v=0\) and \(u=1\). At \((1,0,0)\), one has \(t=0\), \(\gamma=1\), and \(w=1\). At \((-1,0,2)\), since \(k+1\) is even,

\[
t=(-1)^{k+1}2=2,
\qquad
\gamma=-1,
\qquad
w=-1.
\]

For odd \(k,d\), direct substitution gives

\[
q(1)=q(-1)=-1,
\qquad
Q(1)=Q(-1)=0,
\]

and therefore

\[
p(1)=p(-1)=-\frac1{k+1}.
\]

Consequently \(\alpha=\beta=0\) at both points, while \(x\gamma=1\). The source points are distinct, so \(F_{k,d}\) is noninjective. \(\square\)

## 6. Fully expanded degree-eight specialization

For \((k,d)=(2,3)\), let \(a=x^2y\). Then

\[
\begin{aligned}
F_1={}&z(1+a)^4+
\frac{xy^2}{3}(5a^3+17a^2+20a+8),\\
F_2={}&4xz(1+a)^3+
\frac{y}{3}(20a^3+48a^2+33a+2),\\
F_3={}&-x^4z+\frac{x}{3}(3-5a).
\end{aligned}
\]

Exact differentiation gives

\[
\det JF=-\frac23.
\]

The distinct rational points

\[
p_1=\left(\frac83,-\frac9{64},\frac{495}{4096}\right),
\qquad
p_2=\left(\frac83,\frac9{64},-\frac{225}{4096}\right)
\]

satisfy

\[
F(p_1)=F(p_2)=\left(0,\frac9{64},1\right).
\]

## 7. Reproducibility

The companion archive contains one executed exact-arithmetic verifier and two portable independent implementations:

1. `verify_sympy.py` — executed in the supplied environment; emits `certificate_sympy.json` and passes.
2. `verify_sage.py` — an independent Sage polynomial-ring implementation supplied for external execution.
3. `verify_maple.mpl` — an independent Maple implementation supplied for external execution.

An independent ProofKit reproduction also passed all four central claims; its signed ledger is `independent_reproduction.md` (run ID `cbbb82dc-f35e-4c24-aeb0-8d9b2159c95a`).

The executed SymPy verifier and independent reproduction check:

- polynomiality;
- the exact constant Jacobian over a parameter grid;
- the uniform odd-parameter collision;
- the fully expanded \((2,3)\) map;
- exact distinctness of the collision points;
- exact equality of their images.

The theorem proofs, rather than the finite parameter grid, establish the arbitrary-parameter claims.

## 8. Relation to the existing weighted-lift row

The case \(k=1\) lies in the previously described weight class \((1,-1,-2)\). The present construction introduces the independent weight parameter \(k\); its source weights are \((1,-k,-k-1)\), and its geometric generic degree is multiplied by \(k\). The specialization \((k,d)=(2,3)\) recovers the degree-eight specimen coordinate-for-coordinate.

No historical priority claim is made here beyond these exact mathematical comparisons.