Divy Srivastava littledivy

The "-2 Jacobian" non-injective map — what a search turns up

Search query: Jacobian conjecture fake counterexample constant Jacobian -2 non-injective

The hit: Vitushkin's classic example

A map with Jacobian determinant exactly −2 that is not injective is a well-known "trick" example, usually attributed to Vitushkin:

clawpatrol — local dev setup

Prerequisites: Go 1.22+, Node 18+, npm

Step 1 — create dev config

cat > dev.hcl <<'HCL'
listen      = "0.0.0.0:8443"
info_listen = "0.0.0.0:8080"

$ deno sandbox new --expose-http 18789 --memory 3gb --ssh

Created sandbox with id 'sbx_ord_f02azv7t0x7tjxz4y51n'
Exposed port 18789 to https://1ef0cf0c631d49b9b5ad1300ec939252.sandbox.deno.net
ssh 1531786bd2d9b9c96f56095a0f7e2dcd@ssh.deno.net

            __    ╭─────────────────────────────────────╮
           /ᵔ_)  <  Welcome to Deno Sandbox             │
 .-----/ / │ Sandbox sbx_ord_f02azv7t0x7tjxz4y51n│

sh <(curl -L https://nixos.org/nix/install)
mkdir -p ~/.config/nix 
cat <<EOF > ~/.config/nix/nix.conf
experimental-features = nix-command flakes
EOF
cd ~/.config/nix 
nix flake init -t nix-darwin

nix run nix-darwin -- switch --flake .

Migrate from ring to aws_lc_rs

rg -l 'ring::' --glob '*.rs' | xargs -I{} perl -pi -e 's/\bring::/aws_lc_rs::/g' {}
rg -l --fixed-strings 'ring.workspace' --glob '*.toml' \
  | xargs -I{} perl -pi -e 's/\bring\.workspace\b/aws-lc-rs.workspace/g' {}

Icons in PE are very sensitive to resource table ordering. Here's what works:

We can't use editpe's builtin set_icon because it lays out Icon and Icon Group together before/after the RCData. Instead, we need a custom implementation that places RCData between the icon entries. This means we defer Icon Group to always happen during PortableExecutable::build.

After the fix:

	import sympy as sp

	x, y, z = sp.symbols('x y z')
	u = 1 + x*y
	w = 4 + 3xy
	F = [u*3z + y*2uw, y + 3xu2z + 3xy*2w, 2x - 3x*2y - x*3z]

	J = sp.Matrix([[sp.diff(f, v) for v in (x, y, z)] for f in F])
	print(sp.expand(J.det()))

	#!/usr/bin/env python3
	"""
	Verify the claimed properties of the polynomial map F: C^3 -> C^3

	F0 = (1+xy)^3 z + y^2 (1+xy)(4+3xy)
	F1 = y + 3x(1+xy)^2 z + 3x y^2 (4+3xy)
	F2 = 2x - 3x^2 y - x^3 z

	Claims checked:
	1. det(Jacobian F) is the constant -2 (nonzero constant everywhere)

	/-
	Jacobian "counterexample" verifier.

	The map F : ℂ³ → ℂ³ (here checked over ℚ ⊂ ℂ; all data is rational)

	F₀ = (1+xy)³ z + y² (1+xy) (4+3xy)
	F₁ = y + 3x (1+xy)² z + 3x y² (4+3xy)
	F₂ = 2x − 3x² y − x³ z

	Claimed facts, machine-verified below:

	#!/bin/bash
	set -e

	rm -rf gen_temp gen
	mkdir -p gen_temp

	gh run download 21040538462 -p "src_binding_*.rs" -D gen_temp

	mkdir -p gen
	for dir in gen_temp/*/; do