{"uid":"cap_19Oktn4Y1RemZxYNvVqOQ","slug":"hermite-reduction-for-proper-rational-functions-e5904da6","name":"Hermite Reduction for Proper Rational Functions","description":"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.","url":"https://detect.halowerk.com/v1/F-B-NIGHT-FINITE-STRUCTURES/rational-differential-hermite-reduction?utm_source=zero.xyz","method":"POST","headers":{},"bodySchema":{"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}}},"responseSchema":null,"example":null,"exampleRequest":null,"tags":["x402"],"displayCostAmount":"0.003","displayCostAsset":"USDC","priceDynamic":false,"priceHint":null,"priceStatus":"priced","priceSource":"probe","requiresHandshake":false,"reviewCount":0,"rating":{"score":"0.00","successRate":"0.00","reviews":0,"stars":null,"state":"unrated"},"availabilityStatus":"unknown","priceObserved":null,"sessionDeposit":null,"pricing":{"kind":"static","summary":"$0.003/call","primary":{"kind":"static","protocol":"x402","network":"base","amountUsd":"0.003","per":"call","confidence":"exact"},"accepted":[{"kind":"static","protocol":"x402","network":"base","amountUsd":"0.003","per":"call","confidence":"exact"}]},"paymentMethods":[{"uid":"pm_RrLQR32EGvif4QLMCvmVj","protocol":"x402","methodType":"crypto","chain":"base","mode":"charge","costAmount":"0.003","costPer":"request","priority":0,"asset":"0x833589fCD6eDb6E08f4c7C32D4f71b54bdA02913","unit":"request","depositMicros":null,"planRef":null}],"brandName":null,"brandSlug":null,"brandBaseUrl":null,"brandDocsUrl":null,"whatItDoes":"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.","exampleAgentPrompt":"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?","exampleUseCases":[{"title":"CAS pipeline integration preflight","prompt":"Before my symbolic integrator tries to integrate the rational function with numerator [3, 0, -2] and denominator [1, 2, 1, 2, 1], I've claimed the rational part is [1] over [1, 1] and the logarithmic remainder is [2] over [1, 1, 1]. Can you verify whether that Hermite reduction is exactly right?"},{"title":"Automated conformance check for a student CAS","prompt":"My algebra system produced a Hermite reduction for the proper rational function whose numerator is [0, 1, -1] and denominator is [1, 0, 0, 1]. It claims rational part numerator [1] over [1, 1] and remainder numerator [0, 1] over [1, 0, 1]. Please confirm or refute that decomposition."},{"title":"Symbolic integration agent validation step","prompt":"I'm building a symbolic integrator and need to check the Hermite reduction before proceeding. The function has numerator coefficients [2, -1] and denominator [1, 0, 2, 0, 1]. My claimed rational part is [1, 0] over [1, 0, 1] and the squarefree remainder is [1] over [1, 0, 1]. Is this reduction correct?"}],"resultDescription":"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.","failureModes":["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"],"whenToPreferThis":"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.","instructions":null,"reviewSummary":null,"reviewSummaryHighlights":null,"reviewSummaryConcerns":null,"reviewSummaryGeneratedAt":null,"activationCount":0,"lastUsedAt":null,"lastSuccessfullyRanAt":null,"lastHealthCheckAt":"2026-11-02T12:31:12.069Z","isFirstParty":false,"canonicalSlug":"hermite-reduction-for-proper-rational-functions-e5904da6"}