6 ms·
"He proved that if a curve’s equation has a variable raised to a power higher than 3, then it must have a finite number of [rational] points."
by ljsprague 6mo ago
"He proved that if a curve’s equation has a variable raised to a power higher than 3, then it must have a finite number of [rational] points."
- OgsyedIE 6mo agoThis must be an incorrect description of what has actually been proved, since x^4 is a counterexample.
- raphlinus 6mo agoMy understanding, which is to be taken with a grain of salt, is that there's an additional constraint, not stated in the Scientific American article, that the plane curve be irreducible. The example of x^4 is reducible, it's x^2 * x^2 among other thing. The actual conjecture is expressed in terms of genus, but this follows from the genus-degree formula.
- deleted 6mo ago[deleted]
- thornhill 6mo agoThe curve they mean y = x^4 is irreducible but the genus is 0 since it’s isomorphic to the affine line.
- jlev1 6mo agoThe correct description is “a smooth curve of genus at least 2”. https://en.wikipedia.org/wiki/Faltings%27_theorem https://en.wikipedia.org/wiki/Faltings%27_theorem The reason for the confusion is that a smooth, projective plane curve of degree d has genus (d-1)(d-2)/2, which is 2 or greater starting at d=4. Hence the phrasing in the article, which is missing the “smooth, projective” hypothesis. The equation y = x^4 doesn’t define a smooth curve when extended to the projective plane, because it has a singularity at infinity.
- thornhill 6mo agoI think the theorem applies to any curve, if you take geometric genus.