Pith Number

pith:3FGK36MJ

pith:2026:3FGK36MJKOIWI76B5W3VPQFPW5

not attested not anchored not stored refs resolved

Componentwise height bounds for polynomial value-set lifting

Henry Shin

Rational components with one geometric point at infinity contribute sharp power-log order B^{[k:Q]/d_X(C)} (log B)^{|S|-1} when S-active.

arxiv:2605.12903 v1 · 2026-05-13 · math.NT · math.AG

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\pithnumber{3FGK36MJKOIWI76B5W3VPQFPW5}

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Claims

C1strongest claim

For a rational component of f(X)-g(Y)=0 with one geometric point at infinity and projection degree d_X(C) to the X-line, the corresponding contribution has the sharp power-log order B^{[k:Q]/d_X(C)}(log B)^{q_{k,S}}, where q_{k,S}=rk O_{k,S}^* = |S|-1, precisely when its X-parametrization is S-active.

C2weakest assumption

The analysis assumes that after removing the graph components Y=h(X) with f=g o h, the remaining rational components of the curve f(X)-g(Y)=0 can be classified by their number of geometric points at infinity and that the S-activity of the X-parametrization is well-defined and detectable.

C3one line summary

Rational components of f(X)-g(Y)=0 with one geometric point at infinity and projection degree d contribute asymptotically B^{[k:Q]/d} (log B)^{|S|-1} when S-active, while others contribute at most polylogarithmically or finitely many terms.

References

17 extracted · 17 resolved · 0 Pith anchors

[1] P. Alvanos, Y. F. Bilu, and D. Poulakis,Characterizing algebraic curves with in- finitely many integral points, Int. J. Number Theory5(2009), no. 4, 585–590, DOI 10.1142/S1793042109002274 2009 · doi:10.1142/s1793042109002274

[2] R. M. Avanzi and U. M. Zannier,Genus one curves defined by separated variable polynomials and a polynomial Pell equation, Acta Arith.99(2001), no. 3, 227–256, DOI 10.4064/aa99-3-2 2001 · doi:10.4064/aa99-3-2

[3] Barroero,AlgebraicS-integers of fixed degree and bounded height, Acta Arith 2015 · doi:10.4064/aa167-1-4

[4] Y. F. Bilu,Quadratic factors off(x)−g(y), Acta Arith.90(1999), no. 4, 341–355, DOI 10.4064/aa-90-4-341-355 1999 · doi:10.4064/aa-90-4-341-355

[5] Y. F. Bilu and R. F. Tichy,The Diophantine equationf(x) =g(y), Acta Arith.95 (2000), no. 3, 261–288, DOI 10.4064/aa-95-3-261-288 2000 · doi:10.4064/aa-95-3-261-288

Receipt and verification

First computed	2026-05-18T03:09:10.676671Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

d94cadf9895391647fc1edb757c0afb76d3f8012a832c6e8bd7e1664e4f9d0c1

Aliases

arxiv: 2605.12903 · arxiv_version: 2605.12903v1 · doi: 10.48550/arxiv.2605.12903 · pith_short_12: 3FGK36MJKOIW · pith_short_16: 3FGK36MJKOIWI76B · pith_short_8: 3FGK36MJ

Agent API

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/3FGK36MJKOIWI76B5W3VPQFPW5 \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: d94cadf9895391647fc1edb757c0afb76d3f8012a832c6e8bd7e1664e4f9d0c1

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "ed76e1982b3003ba17d7721bdca813c4e3916995d931b1c37684c06cfd049b8c",
    "cross_cats_sorted": [
      "math.AG"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.NT",
    "submitted_at": "2026-05-13T02:28:01Z",
    "title_canon_sha256": "a9480d13dce943905f9a7b0f18d5ce84e3ec4845c00316fddf5933f42a9bf6ea"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.12903",
    "kind": "arxiv",
    "version": 1
  }
}