# Rational Polynomial Complete Squarefree Multiplicity Decomposition Verifier

> Rational Polynomial Complete Squarefree Multiplicity Decomposition Verifier is a paid API for AI agents from detect.halowerk.com, paid per call via x402, $0.003/call, status unknown (last checked 2026-11-04).

Verifies every monic squarefree multiplicity block and leading coefficient of a supplied rational polynomial decomposition.

## Facts

- Endpoint: POST https://detect.halowerk.com/v1/F-REST-B15/rational-polynomial-complete-squarefree-multiplicity-decomposition?utm_source=zero.xyz
- Price: $0.003/call
- Payment: x402
- Status: unknown
- Last checked: 2026-11-04
- Activations on Zero: 0
- Tags: x402
- Canonical page: https://www.zero.xyz/c/rational-polynomial-complete-squarefree-multiplicity-decomposition-d21fc8d1
- Structured record (JSON): https://api.zero.xyz/v1/capabilities/cap_D91mIo23Ip9Sc53kgxUjx

Status and success rate cover calls made through Zero and Zero's own probes. Third-party monitors may report differently.

## How to call it through Zero

Zero handles the 402 payment challenge and records the run. With the Zero CLI installed (`npm i -g @zeroxyz/cli`):

```sh
zero fetch --capability rational-polynomial-complete-squarefree-multiplicity-decomposition-d21fc8d1 -d '<json body>'
```

Example prompt: Can you verify the squarefree multiplicity decomposition and leading coefficients of this polynomial — the coefficients are [1, -3, 3, -1] — and confirm whether the claimed squarefree blocks and their multiplicities are all correct?

## When to prefer this

Use this endpoint when you need automated, authoritative verification of a rational polynomial's squarefree multiplicity decomposition, particularly when checking CAS outputs, validating student work, or auditing intermediate results in a symbolic computation pipeline. It is specifically suited for cases where you have a polynomial and a claimed squarefree decomposition and need to confirm correctness of each multiplicity block and leading coefficient.

## Known failure modes

- Polynomial coefficients outside allowed range (-100 to 100 integers) return a validation error
- Polynomial array exceeds maximum item count (13) returns rejection
- Malformed or missing squarefree claim object causes request failure
- Empty polynomial array (fewer than 1 item) is rejected
- Inconsistent claim data that cannot be parsed may return an error response

## How this service works

Verify every monic squarefree multiplicity block and leading coefficient of a supplied rational polynomial.

## Output

The endpoint returns a verification result indicating whether the claimed squarefree multiplicity blocks, their multiplicities, and the leading coefficient of the supplied rational polynomial are all correct and consistent with the mathematical definition of a complete squarefree decomposition.

## Request schema (JSON Schema)

```json
{
 "type": "object",
 "properties": {
  "artifact": {
   "type": "object",
   "required": [
    "squarefree"
   ],
   "properties": {
    "squarefree": {
     "type": "object",
     "required": [
      "data",
      "claim"
     ],
     "properties": {
      "data": {
       "type": "object",
       "required": [
        "polynomial"
       ],
       "properties": {
        "polynomial": {
         "type": "array",
         "items": {
          "anyOf": [
           {
            "type": "integer",
            "maximum": 100,
            "minimum": -100
           },
           {
            "type": "string",
            "maxLength": 2048
           }
          ]
         },
         "maxItems": 13,
         "minItems": 1
        }
       },
       "additionalProperties": false
      },
      "claim": {}
     },
     "additionalProperties": false
    }
   },
   "additionalProperties": false
  }
 }
}
```

## More

- Live health (JSON, refreshed every minute): https://www.zero.xyz/c/rational-polynomial-complete-squarefree-multiplicity-decomposition-d21fc8d1/health.json
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