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The formal derivative of a polynomial
Definition
Let be a commutative ring and let . Its formal derivative is
Here means the sum of copies of in the additive group of . Equivalently, the coefficient of in is . The sequence defining has finite support because the coefficient sequence of does (The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution). This is an algebraic operation and does not use a limit.
Depends on
Used by
- Certified complex-polynomial root isolation in exact algebraic arithmetic Algorithm
- Real bounded-polynomial QSP phase synthesis Algorithm
- For monic f of degree n, Res(f,f')=(-1)^n(n-1)/2Disc(f) Corollary
- The A₂ discriminant, its Jacobian and the top coinvariant class in ℂ[u,z]/(uz,u³+z³) Example
- The exceptional spectra for E₆ and H₃ computed exactly: characteristic polynomials, cyclotomic factorisations and the resulting degree tables Example
- The invariants and the coinvariant Hilbert series of I₂(m): an explicit computation and the noncrystallographic contrast Example
- A distribution with a vanishing derivative is a polynomial Lemma
- Algebraicity of the coordinates over the invariant field and non-vanishing of the invariant Jacobian Lemma
- An explicit root-separation bound for squarefree integer polynomials Lemma
- Continuous Markov derivative bound on an interval Lemma
- Exact gcd squarefree decomposition preserves roots and multiplicities Lemma
- Exact interval boundedness by multiplicity strata and Sturm counts Lemma
- In characteristic p, the roots of x^pⁿ-x form a subfield and are all simple Lemma
- Polynomial convolutions force activation polynomiality almost everywhere Lemma
- Sturm signed remainders count distinct real polynomial roots Lemma
- The six exceptional Coxeter spectra: characteristic polynomials, orders and spectral exponents from exact matrices Lemma
- Linearity, power rule, Leibniz rule and the degree bound for the formal derivative Proposition
- Correctness and termination of certified complex-polynomial root isolation Theorem
- Correctness of QSP phase synthesis Theorem
- Newton's identities: k eₖ=∑ᵢ₌₁ᵏ(-1)ⁱ⁻¹eₖ₋ᵢpᵢ Theorem
- Singular locus of a reduced analytic hypersurface Theorem
- Sturm sign variation counts distinct real polynomial roots Theorem
- The total degree sum, the invariant Jacobian as the discriminant, anti-invariants, and the top coinvariant class Theorem
- tⁿ-1 is separable over K exactly when the characteristic does not divide n, and then a splitting field carries n distinct n-th roots of unity Theorem
Dependency tree · two levels
3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Brian Conrad, Differential Criterion and Primitivity, Section 1 (standard reference, not scraped)