Description
Return leading coefficient under affine input substitution.
Polynomial transformations change coefficient scales or root coordinates under an explicit substitution. Translation, reflection and dilation use different signs, while a single complex phase does not apply a complete root-of-unity filter. The original function’s domain still matters after rational cancellation.
Return leading coefficient under affine input substitution.
When to use this tool
Use Polynomial Affine Substitution Leading Coefficient when checking the stated polynomial transformation or complex-coordinate term.
How to use the tool and how it works
- Enter leading, degree, scale. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read leading, degree. Use the output convention below.
Return leading coefficient under affine input substitution. Require nonzero leading/scale and whole degree 0..128; returned leading = leading*scale^degree, degree is preserved.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| leading | Leading | Required. Finite number. |
| degree | Degree | Required. Safe integer; minimum 0; maximum 128. |
| scale | Nonzero signed multiplier in p(scale*x); leading changes by scale^degree. | Required. Finite number. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| leading | Leading | Required. Finite number. |
| degree | Degree | Required. Safe integer. |
{
"input": {
"leading": 3,
"degree": 2,
"scale": 2
},
"output": {
"leading": 12,
"degree": 2
}
}Limitations and assumptions
- The affine shift does not affect the leading term; actual degree and the remaining transformed coefficients are not verified.
- Inputs must satisfy both the field types and the cross-field conditions above. Invalid input causes an error; a successful false flag or documented placeholder is a different result.
- Numeric calculations use binary floating point unless exact decimal-integer arithmetic is explicitly stated. No hidden measurement tolerance or decimal rounding is added; nonfinite or out-of-contract results are rejected. Numeric signed-zero results are exported as zero.
Alternative or Complementary methods
Use the full transformed expression and explicit variable substitution for complete symbolic equivalence, filtering or root analysis.
References
Similar or alternative tools
- Polynomial Coefficient Binomial Shift Term
Return one binomial-shift coefficient contribution.
- Polynomial Affine Input Shift Root Coordinate
Transform a root coordinate under affine input substitution.