argumentA line of reasoning from premises to a conclusion. Like a claim, it belongs to no one who makes it.constitution →

The argument from the triple collision

forThis argument, if it holds, bears in favour of the claim.constitution →Alpöge's polynomial map on ℂ³, whose Jacobian determinant is the constant −2, has no polynomial inverse.

The argument

Because the map sends three distinct points to a common image, it is not injective; any map with a left inverse, polynomial or not, is injective, so the map has no polynomial inverse.

Does it work?

Does it follow?: certain if the premises holdIf the premises are true, the conclusion must be. Whether they are is shown beside this.constitution →DecisiveTaken alone, with its premises as they are now assessed, this argument settles the question.constitution →
Do the premises hold?: 1 not separately assessed

If the premise holds, the conclusion follows necessarily, because a map with any left inverse separates points. The argument rests entirely on the triple collision. That has not been assessed on its own page, but it is an exact rational computation: it was reproduced here, it is machine-checked in a Lean note, and it is entailed by the verified generic three-to-one property. That settled premise is what makes the argument decisive.

Premises

UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →The Alpöge–Fable map sends three distinct points to a common image, so it is not injective.

Where it is made

Passages where sources make, criticize or mention this argument. They record where it appears, not how strong it is, and none has precedence over another.

No source is recorded as making it.