Description
Count polynomial coefficient sign variations ignoring exact zeros in ascending order; this is algebraic evidence, not an exact positive-root count.
A coefficient list here is constant-first: index j multiplies x to power j. Thus [1, 2, 3] represents 1 + 2x + 3x². Trailing list zeros are high-power zeros; stored length can differ from mathematical degree, and the zero polynomial needs a separate degree convention.
Count polynomial coefficient sign variations ignoring exact zeros in ascending order; this is algebraic evidence, not an exact positive-root count.
When to use this tool
Use this tool to count polynomial coefficient sign variations ignoring exact zeros in ascending order; this is algebraic evidence, not an exact positive-root count.
How to use the tool and how it works
- Enter coefficients using the field definitions below.
- Run the operation. Correct input errors before interpreting an output.
- Read result.variations. Use the ordering, boundary rules and absence conditions described below.
Discard exactly zero coefficients and count sign changes between consecutive remaining coefficients.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| coefficients | Nonempty coefficients in ascending power order [constant, x, x², ...]. | Dense list of at most 128 finite numbers; integer/shape constraints above also apply. |
| Output field | Interpretation |
|---|---|
| result | Object containing the named fields below. |
| result.variations | Sign-change count after removing exactly zero coefficients. |
{
"input": {
"coefficients": [
1,
0,
-2,
3
]
},
"output": {
"result": {
"variations": 2
}
}
}Limitations and assumptions
- The result is a sign-variation count, not an exact root count or a root solver.
- Coefficient lists use ascending powers and ordinary floating-point arithmetic. Nonempty input is required; individual operations impose tighter size limits. The tool does not attach or check physical units for x and its coefficients.
- Every declared top-level input field is required and unexpected fields are rejected. Ordinary strings are limited to 8,192 UTF-16 code units and lists to 128 entries, with tighter operation-specific bounds where stated. JSON text has bounded nesting and node counts.
- Numeric outputs must be finite and no larger in absolute magnitude than the safe-integer limit; operations returning exact large integers use decimal text. Positions are zero-based and spans end exclusively unless the operation explicitly states a different convention.
Alternative or Complementary methods
Check coefficient order before importing data from another algebra library. Use separate root-finding or symbolic algebra methods when an operation requires more than the stated coefficient transformation.
References
Similar or alternative tools
- Polynomial Antiderivative Coefficient List
Integrate ascending-power polynomial coefficients with an explicitly supplied integration constant; bounded number arithmetic, no symbolic rational simplification.
- Polynomial Coefficient Convolution Product
Multiply two ascending-power polynomial coefficient lists by convolution and normalize high-power exact zeros; no FFT or symbolic integer arithmetic.