# Hermite Reduction for Proper Rational Functions

> Hermite Reduction for Proper Rational Functions is a paid API for AI agents from detect.halowerk.com, paid per call via x402, $0.003/call, status unknown (last checked 2026-11-02).

Verifies and returns the exact rational part and squarefree logarithmic remainder of the Hermite reduction of a proper rational function, given numerator, denominator, and a candidate claim.

## Facts

- Endpoint: POST https://detect.halowerk.com/v1/F-B-NIGHT-FINITE-STRUCTURES/rational-differential-hermite-reduction?utm_source=zero.xyz
- Price: $0.003/call
- Payment: x402
- Status: unknown
- Last checked: 2026-11-02
- Activations on Zero: 0
- Tags: x402
- Canonical page: https://www.zero.xyz/c/hermite-reduction-for-proper-rational-functions-e5904da6
- Structured record (JSON): https://api.zero.xyz/v1/capabilities/cap_19Oktn4Y1RemZxYNvVqOQ

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 hermite-reduction-for-proper-rational-functions-e5904da6 -d '<json body>'
```

Example prompt: I need to verify the Hermite reduction of the rational function with numerator coefficients [1, 0, -1] and denominator coefficients [1, 0, 2, 0, 1]. The claimed rational part is numerator [0, 1] over denominator [1, 0, 1], and the logarithmic remainder is numerator [1] over denominator [1, 0, 1]. Can you check whether that decomposition is exactly correct?

## When to prefer this

Use this endpoint when a symbolic integration agent needs to verify or obtain the exact Hermite reduction of a proper rational function before proceeding to the logarithmic integration step. It is specifically designed for conformance checking of Hermite reduction claims, not generic GCD or polynomial gradient operations. Prefer this over general CAS services when you need a fast, pay-per-call verification step in an automated integration pipeline with exact rational arithmetic.

## Known failure modes

- Numerator degree not strictly less than denominator degree (not a proper rational function) — returns validation error
- Denominator has fewer than 2 coefficients — returns schema error
- Coefficient values outside allowed integer range [-64, 64] — returns schema error
- Claim polynomial arrays contain non-rational-number strings — returns pattern mismatch error
- Claim does not match the correct Hermite reduction — returns conformance failure with correct values
- Malformed artifact object missing required fields — returns 400/422

## How this service works

A symbolic-integration conformance agent needs the full exact rational derivative part and squarefree logarithmicremainder before integratinga proper rational function. GenericGCDdegree or polynomialgradient cannotproduce this reduction.

## Output

Returns confirmation or correction of the Hermite reduction, including the exact rational derivative part (rational numerator and denominator polynomials) and the squarefree logarithmic remainder (remainder numerator and denominator), enabling a symbolic integration agent to proceed with the logarithmic integration step.

## Request schema (JSON Schema)

```json
{
 "type": "object",
 "properties": {
  "artifact": {
   "type": "object",
   "required": [
    "hermite_integral"
   ],
   "properties": {
    "hermite_integral": {
     "type": "object",
     "required": [
      "numerator",
      "denominator",
      "claim"
     ],
     "properties": {
      "claim": {
       "type": "object",
       "required": [
        "rational_numerator",
        "rational_denominator",
        "remainder_numerator",
        "remainder_denominator"
       ],
       "properties": {
        "rational_numerator": {
         "type": "array",
         "items": {
          "type": "string",
          "pattern": "^-?(0|[1-9][0-9]*)(/[1-9][0-9]*)?$",
          "maxLength": 160
         },
         "maxItems": 9,
         "minItems": 1
        },
        "remainder_numerator": {
         "type": "array",
         "items": {
          "type": "string",
          "pattern": "^-?(0|[1-9][0-9]*)(/[1-9][0-9]*)?$",
          "maxLength": 160
         },
         "maxItems": 9,
         "minItems": 1
        },
        "rational_denominator": {
         "type": "array",
         "items": {
          "type": "string",
          "pattern": "^-?(0|[1-9][0-9]*)(/[1-9][0-9]*)?$",
          "maxLength": 160
         },
         "maxItems": 9,
         "minItems": 1
        },
        "remainder_denominator": {
         "type": "array",
         "items": {
          "type": "string",
          "pattern": "^-?(0|[1-9][0-9]*)(/[1-9][0-9]*)?$",
          "maxLength": 160
         },
         "maxItems": 9,
         "minItems": 1
        }
       },
       "additionalProperties": false
      },
      "numerator": {
       "type": "array",
       "items": {
        "type": "integer",
        "maximum": 64,
        "minimum": -64
       },
       "maxItems": 8,
       "minItems": 0
      },
      "denominator": {
       "type": "array",
       "items": {
        "type": "integer",
        "maximum": 64,
        "minimum": -64
       },
       "maxItems": 9,
       "minItems": 2
      }
     },
     "additionalProperties": false
    }
   },
   "additionalProperties": false
  }
 }
}
```

## More

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