Pith Number

pith:BMMOO7NN

pith:2026:BMMOO7NNT5OQOUMJQCAULN557V

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G{\aa}rding Polynomials

Biao Ma, Hao Fang

Gårding polynomials are defined by positivity regions invariant under positive translations and positive affine maps, yielding two characterizations that extend real stable polynomials while preserving Rayleigh and log-concavity properties.

arxiv:2604.27755 v2 · 2026-04-30 · math.CO · math.AP · math.CA · math.OC · math.PR

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Claims

C1strongest claim

We establish a structural theorem providing two complementary characterizations of this class: one via reduction to the multi-affine case through polarization, and another via a recursive condition involving partial derivatives. The class of Gårding polynomials strictly extends that of real stable polynomials while retaining many of their structural properties. In particular, multi-affine Gårding polynomials with nonnegative coefficients satisfy the Rayleigh property, and their positive univariate specializations yield ultra log-concave coefficient sequences. Moreover, the Gårding property for several matroid generating functions is preserved under natural matroid operations.

C2weakest assumption

The defining assumption that the positivity regions are invariant under translation by positive directions and closed under strictly positive affine transformations is sufficient to guarantee the two characterizations and the preservation of the Gårding property under matroid operations.

C3one line summary

Gårding polynomials extend real stable polynomials via new characterizations through polarization and partial derivatives, preserving key properties under matroid operations and enabling broader negative dependence proofs.

Receipt and verification

First computed	2026-05-20T00:05:45.658969Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

0b18e77dad9f5d075189808145b7bdfd5eeaf6756004c7a81600ef1d29750701

Aliases

arxiv: 2604.27755 · arxiv_version: 2604.27755v2 · doi: 10.48550/arxiv.2604.27755 · pith_short_12: BMMOO7NNT5OQ · pith_short_16: BMMOO7NNT5OQOUMJ · pith_short_8: BMMOO7NN

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/BMMOO7NNT5OQOUMJQCAULN557V \
  | 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: 0b18e77dad9f5d075189808145b7bdfd5eeaf6756004c7a81600ef1d29750701

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "55c7b58acff17d52fd7bd0730744ebf449e12fb8d473ab3895e532ec539501e1",
    "cross_cats_sorted": [
      "math.AP",
      "math.CA",
      "math.OC",
      "math.PR"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.CO",
    "submitted_at": "2026-04-30T11:43:39Z",
    "title_canon_sha256": "c0fa38d6b40d9946e87dd253fb52113594a276c6f84a2ad8d7ad65a3d099e871"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2604.27755",
    "kind": "arxiv",
    "version": 2
  }
}