Pith Number

pith:QVBON24L

pith:2026:QVBON24LQJG2YYLU7IER5H6JGO

not attested not anchored not stored refs resolved

Commutative Semifields from bijections of the Desarguesian plane

Faruk G\"olo\u{g}lu, Lukas K\"olsch

Semiquadratic homogeneous bijections of the Desarguesian plane produce large families of commutative semifields that are neither fields nor twisted fields.

arxiv:2605.14009 v1 · 2026-05-13 · math.CO · math.AC

Open paper page JSON Open Graph Bundle Merged state Verified badge What is a Pith Number?

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{QVBON24LQJG2YYLU7IER5H6JGO}

Prints a linked badge after your title and injects PDF metadata. Compiles on arXiv. Learn more · Embed verified badge

Record completeness

1 Bitcoin timestamp

2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

✓ Portable graph bundle live · download bundle · merged state

The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

we give a large class of semiquadratic homogeneous bijections of P^2(F_q) that are inequivalent to Dembowski-Ostrom monomials. Using these bijections, we construct a large family of commutative semifields that are non-isotopic to finite fields or twisted fields, which in turn give rise to a large family of non-Desarguesian commutative semifield planes.

C2weakest assumption

The constructed maps are indeed bijections and the resulting multiplication defines a semifield (i.e., the algebraic identities hold for the chosen parameters), which must be verified explicitly for each member of the family.

C3one line summary

A large family of commutative semifields non-isotopic to fields or Albert twisted fields is obtained from new semiquadratic bijections on P^2(F_q).

References

25 extracted · 25 resolved · 0 Pith anchors

[1] Shreeram Abhyankar,Projective polynomials, Proceedings of the American Mathematical Society125(1997), no. 6, 1643–1650 1997

[2] A. A. Albert,Finite division algebras and finite planes, Proc. Sympos. Appl. Math., Vol. 10, American Math- ematical Society, Providence, R.I., 1960, pp. 53–70. MR 0116036 1960

[3] J¨ urgen Bierbrauer,New semifields, PN and APN functions, Des. Codes Cryptogr.54(2010), no. 3, 189–200. MR 2584973 2010

[4] Mauro Biliotti, Vikram Jha, and Norman L Johnson,The collineation groups of generalized twisted field planes, Geometriae Dedicata76(1999), 97–126 1999

[5] Coulter and Marie Henderson,Commutative presemifields and semifields, Adv 2008

Cited by

1 paper in Pith

Triprojective almost perfect nonlinear permutations and functions

Receipt and verification

First computed	2026-05-17T23:39:13.081244Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

8542e6eb8b824dac6174fa091e9fc9338a481248add9fe394014bb3f890c1d37

Aliases

arxiv: 2605.14009 · arxiv_version: 2605.14009v1 · doi: 10.48550/arxiv.2605.14009 · pith_short_12: QVBON24LQJG2 · pith_short_16: QVBON24LQJG2YYLU · pith_short_8: QVBON24L

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/QVBON24LQJG2YYLU7IER5H6JGO \
  | 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: 8542e6eb8b824dac6174fa091e9fc9338a481248add9fe394014bb3f890c1d37

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "38aa49f9ac425c96fdb376ee18f589c4e0a5f5c3b7bb427d128181d06e4479e8",
    "cross_cats_sorted": [
      "math.AC"
    ],
    "license": "http://creativecommons.org/publicdomain/zero/1.0/",
    "primary_cat": "math.CO",
    "submitted_at": "2026-05-13T18:14:07Z",
    "title_canon_sha256": "6a3862adcac3ac0329f28a76ab2c1d0cd21dc0b504ee1cbd7c2afc9f0d3dd4a9"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.14009",
    "kind": "arxiv",
    "version": 1
  }
}