Pith Number

pith:ERKW36F7

pith:2026:ERKW36F7KZZCIH5OFWNLX57GHN

not attested not anchored not stored refs pending

On the Principal Minor Expansion and Complexity of the Symmetrized Determinant

Markus Bl\"aser, Mridul Gupta, Sanyam Agarwal

The symmetrized determinant is #P-hard to compute over some polynomial-sized algebras and gives rise to VNP-complete polynomial families.

arxiv:2604.28019 v2 · 2026-04-30 · cs.CC

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

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{ERKW36F7KZZCIH5OFWNLX57GHN}

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 prove that there exists a polynomial-sized algebra such that computing the symmetrized determinant is #P-hard. [...] the associated polynomial family is VNP-complete over a suitable polynomial-dimensional algebra in the non-commutative setting. Further, when seen as a family of polynomials over the matrix algebra, it is also VNP-complete in the commutative setting.

C2weakest assumption

The reductions from known #P-hard and VNP-complete problems to the symmetrized determinant are valid for the specific polynomial-sized associative algebras constructed in the paper, without hidden restrictions on the algebra that would invalidate the hardness for general cases.

C3one line summary

The symmetrized determinant is #P-hard over polynomial-sized algebras and its polynomial family is VNP-complete in both non-commutative and commutative matrix algebra settings.

Receipt and verification

First computed	2026-06-25T01:17:53.599150Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

24556df8bf5672241fae2d9abbf7e63b666f6e14e07ebdb6968b4209027b27cc

Aliases

arxiv: 2604.28019 · arxiv_version: 2604.28019v2 · doi: 10.48550/arxiv.2604.28019 · pith_short_12: ERKW36F7KZZC · pith_short_16: ERKW36F7KZZCIH5O · pith_short_8: ERKW36F7

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/ERKW36F7KZZCIH5OFWNLX57GHN \
  | 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: 24556df8bf5672241fae2d9abbf7e63b666f6e14e07ebdb6968b4209027b27cc

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "18e46183c03f5764e09c5147296a4c64195a810df4460163e849dbf4866eab1c",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "cs.CC",
    "submitted_at": "2026-04-30T15:38:51Z",
    "title_canon_sha256": "4924ea922cdf90a05b37f7d37b6d06386e138bd171d66f9744b59153747bd616"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2604.28019",
    "kind": "arxiv",
    "version": 2
  }
}