Sylvester Resultant Calculator
Compute exact integer resultant by fraction-free Sylvester determinant for constant-first integer polynomials of degree ≤12; zero polynomial returns zero resultant. Nonzero constants have no common roots.
Description
The resultant detects algebraic common-root structure through a Sylvester determinant.
The resultant detects algebraic common-root structure through a Sylvester determinant. Exact integer coefficient arithmetic prevents floating-point near-zero results from being mistaken for exact zeros.1
Compute exact integer resultant by fraction-free Sylvester determinant for constant-first integer polynomials of degree ≤12; zero polynomial returns zero resultant. Nonzero constants have no common roots.
When to use Sylvester Resultant Calculator
Use this tool to compute exact integer resultant by fraction-free Sylvester determinant for constant-first integer polynomials of degree ≤12; zero polynomial returns zero resultant. Nonzero constants have no common roots.
Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
How to use Sylvester Resultant Calculator and understand its results
- Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
- Enter leftPolynomial, rightPolynomial using the field table and calculation rules, then run the tool.
- Read exactSylvesterResultant, commonRootOverAlgebraicClosure. The explanation below defines the units, signs, ordering and any flags or placeholder values.
- If the tool reports an input issue, correct the values and run it again. Apply the stated assumptions when interpreting a successfully returned result.
| Input field | Meaning | Value type; see rules below | Requirement |
|---|---|---|---|
| leftPolynomial | Left polynomial | list of number | Required |
| rightPolynomial | Right polynomial | list of number | Required |
Supply nonempty constant-first integer coefficient lists of at most 13 entries, each bounded in absolute value by one million. Trim high-degree zero coefficients and use fraction-free BigInt elimination. Return the resultant as a decimal string and the common-root flag; zero polynomials give resultant zero, while a nonzero constant has no common root.
resultant = determinant of Sylvester matrix(left polynomial,right polynomial) Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.
| Output field | Meaning | Value type |
|---|---|---|
| exactSylvesterResultant | Exact sylvester resultant | string |
| commonRootOverAlgebraicClosure | Common root over algebraic closure | boolean |
{
"input": {
"leftPolynomial": [
-1,
1
],
"rightPolynomial": [
-2,
1
]
},
"output": {
"exactSylvesterResultant": "-1",
"commonRootOverAlgebraicClosure": false
}
}Limitations and assumptions
Input degree is at most 12. The flag does not locate roots or classify them as real.
Numeric inputs and returned numeric values must be finite; an unrepresentable returned value produces an error. Numeric lists are limited to 100,000 entries; the rules above may require smaller limits. Missing required fields, extra fields, wrong value types and violations of the calculation rules are input errors. A successfully returned false flag is an evaluated outcome; its meaning is explained above.
Alternative or Complementary methods
Use Discriminant Resultant Calculator: Compute exact degree-general integer polynomial discriminant from signed derivative resultant divided by leading coefficient; constant-first degree 1–12, no separate cubic/quartic aliases or real-root-count inference. For a complementary task, use Polynomial GCD Calculator: Exact polynomial GCD over rational coefficients from bounded integer input, normalized monic; zero result is ["0"], rational coefficients are exact decimal numerator/denominator strings.
References
-
Sylvester Resultant Calculator — tool.io
Similar or alternative tools
- Discriminant Resultant Calculator
Compute exact degree-general integer polynomial discriminant from signed derivative resultant divided by leading coefficient; constant-first degree 1–12, no separate cubic/quartic aliases or real-root-count inference.
- Polynomial GCD Calculator
Exact polynomial GCD over rational coefficients from bounded integer input, normalized monic; zero result is ["0"], rational coefficients are exact decimal numerator/denominator strings.