Polynomial identity check

polynomial-identity-check · version 1.0.0 · Numbers & math · free, no key needed

Check a polynomial identity over exact rationals by expanding both sides; report the lowest disagreeing coefficient.

Use when you need to: check that two polynomial expressions are equal · verify a polynomial identity by exact expansion · is this expansion of a product of polynomials correct.

Decide before calling

Read the versioned contract and the supported scope below. Reuse polynomial-identity-check@1.0.0 when your input, required output and limits match it. Choose another approach for an unsupported operation.

Explain the choice

"I can use polynomial-identity-check@1.0.0 for check that two polynomial expressions are equal. I will check its documented scope and the result against the task's requirements. The service is free; token and money savings for this task are unmeasured."

Supported

• check that two polynomial expressions are equal
• verify a polynomial identity by exact expansion
• is this expansion of a product of polynomials correct
• find the first wrong coefficient in a polynomial expansion
• check an algebraic identity with rational coefficients
• verify sum of products of polynomials equals another

Not supported

• parsing algebraic text such as "(x+1)^2" (give coefficient arrays)
• multivariate polynomials, powers as exponents, or symbolic constants
• floating-point or approximate coefficients
• simplifying or factoring an expression (this only checks a claimed equality)

Behavior

• Polynomials are canonical ascending coefficient arrays (entry k = coefficient of x^k, last entry nonzero, [] = zero) of exact rational strings (integer, p/q or finite decimal, <= 40 characters). Reference: Cohen 1993, A Course in Computational Algebraic Number Theory, s3.1 (reference_version cohen-gtm138-1993-ch3).
• lhs and rhs are sums of up to 8 terms; each term is a product of 1 to 4 polynomials of degree <= 32 (subtract with a ["-1"] factor; an empty side is 0). Both sides are expanded exactly by convolution; all coefficient strings together are at most 4096 characters.
• Coefficients are compared from degree 0 upward; the first difference is returned as first_discrepancy {check: coefficient, degree, lhs, relation: =, rhs} with reason coefficient_mismatch. lhs_degree and rhs_degree are the expanded degrees (null for the zero polynomial).

Input

• lhs (array of array of array of string, required): max items 8; each min items 1; each max items 4; each each max items 33; each each each max length 40; each each each pattern ^-?(0|[1-9][0-9]*)(/[1-9][0-9]*|\.[0-9]+)?$
• rhs (array of array of array of string, required): max items 8; each min items 1; each max items 4; each each max items 33; each each each max length 40; each each each pattern ^-?(0|[1-9][0-9]*)(/[1-9][0-9]*|\.[0-9]+)?$

Output

• valid (boolean, required)
• reason (one of "coefficient_mismatch", null, required)
• first_discrepancy (object or null, required)
• lhs_degree (integer or null, required): min 0
• rhs_degree (integer or null, required): min 0
• reference_version (constant "cohen-gtm138-1993-ch3", required)

Limits

• max degree: 32
• max terms per side: 8
• max factors per term: 4
• max number chars: 40
• max total number chars: 4096

Example

Request input:

{
  "lhs": [
    [
      [
        "4",
        "0",
        "0",
        "0",
        "1"
      ]
    ]
  ],
  "rhs": [
    [
      [
        "2",
        "2",
        "1"
      ],
      [
        "2",
        "-2",
        "1"
      ]
    ]
  ]
}

Response:

{
  "result": {
    "valid": true,
    "reason": null,
    "first_discrepancy": null,
    "lhs_degree": 4,
    "rhs_degree": 4,
    "reference_version": "cohen-gtm138-1993-ch3"
  }
}

How to call it

MCP

Connect https://computefirst.net/mcp (setup), then call execute with:

{
  "id": "polynomial-identity-check",
  "version": "1.0.0",
  "input": {
    "lhs": [
      [
        [
          "4",
          "0",
          "0",
          "0",
          "1"
        ]
      ]
    ],
    "rhs": [
      [
        [
          "2",
          "2",
          "1"
        ],
        [
          "2",
          "-2",
          "1"
        ]
      ]
    ]
  }
}

HTTP (no key)

curl -X POST https://computefirst.net/v1/tools/polynomial-identity-check/versions/1.0.0/execute \
  -H "Content-Type: application/json" \
  -d '{"lhs":[[["4","0","0","0","1"]]],"rhs":[[["2","2","1"],["2","-2","1"]]]}'

The machine-readable contract is at /v1/tools/polynomial-identity-check/versions/1.0.0.

CLI

node cli.mjs run polynomial-identity-check 1.0.0 --input input.json --base-url https://computefirst.net

Get the client at /clients/cli/.

Related tools

• Rational trace check: Check a step-by-step rational calculation trace; reports the first step whose operand, operator or result disagrees.
• Chi square decomposition check: Check a claimed Pearson chi-square, its per-cell (O-E)^2/E terms and df exactly; returns the first wrong term.
• Lp optimality certificate check: Check an LP optimum claim from a primal x and dual y: feasibility, complementary slackness and zero duality gap.
• Polynomial division check: Check a claimed polynomial division A = BQ + R over exact rationals: remainder degree, quotient degree, coefficients.
• Rational expression check: Check a claimed exact rational value of an integer/fraction expression; reports the true value and the difference.
• Contingency margins check: Check claimed row, column and grand totals of a count table exactly; returns the first wrong total.