Pith Number

pith:QPHXACAB

pith:2025:QPHXACABVIH5BN42YGPZ7JNFKX

not attested not anchored not stored refs pending

On solutions of singular Sylvester equations in quaternions

Christian Scheunert, Frank H. P. Fitzek, Hristina Radak

Solutions to the singular Sylvester equations ax - xb = 0 and ax - xb = c exist in quaternions under specific conditions and are constructed using quaternion square roots.

arxiv:2510.04629 v7 · 2025-10-06 · math.RA

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\usepackage{pith}
\pithnumber{QPHXACABVIH5BN42YGPZ7JNFKX}

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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

Conditions for the existence of solutions are provided. In addition, the general and nonzero solutions to these equations are derived applying quaternion square roots.

C2weakest assumption

The derivations assume that quaternion square roots can be taken in a manner that preserves the required algebraic relations for the singular case; the paper does not state whether this construction is always possible or requires additional restrictions on a and b.

C3one line summary

Derives existence conditions and explicit general solutions for homogeneous and inhomogeneous singular Sylvester equations over the quaternions using square-root constructions.

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-06-08T01:05:03.053785Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

83cf700801aa0fd0b79ac19f9fa5a555f9e6b2e355df5928954444b56ec74d8b

Aliases

arxiv: 2510.04629 · arxiv_version: 2510.04629v7 · doi: 10.48550/arxiv.2510.04629 · pith_short_12: QPHXACABVIH5 · pith_short_16: QPHXACABVIH5BN42 · pith_short_8: QPHXACAB

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/QPHXACABVIH5BN42YGPZ7JNFKX \
  | 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: 83cf700801aa0fd0b79ac19f9fa5a555f9e6b2e355df5928954444b56ec74d8b

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "d6975e5dfbe301a358a84471e3fd3156ddda7388cf079dbde3d6e865233f3d31",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.RA",
    "submitted_at": "2025-10-06T09:33:39Z",
    "title_canon_sha256": "4c8df032ee18d48c39522532b24317fa839fd27fa5d858f864997139006ef914"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2510.04629",
    "kind": "arxiv",
    "version": 7
  }
}