Evaluation and roots of a polynomial in a commutative target ring

Depends on

• The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution
• Ring homomorphism: additive, multiplicative, and required to send $1$ to $1$

Used by

• Every finite subgroup of the unit group of an integral domain is cyclic Corollary
• Factor theorem over a commutative ring Corollary
• For monic f, if g-g₁=qf, then Res(f,g)=Res(f,g₁) Corollary
• Global functions on geometrically connected and geometrically reduced proper schemes Corollary
• Morse Euler characteristic identity Corollary
• R[x] is Noetherian if and only if R is Noetherian Corollary
• Bezout needs algebraic closure: an imaginary conic has no real point Counterexample
• Over Fₚ, xᵖ-x and the zero polynomial induce the same function but are distinct polynomials Counterexample
• P(x)=x(x-1) vanishes on {0,1} although degₓ P=|{0,1}| Counterexample
• Algebraic and transcendental elements and algebraic extensions Definition
• An algebraically closed field: every nonconstant polynomial has a root in the field Definition
• Character polynomials in the cycle counts Xᵢ Definition
• Classical affine algebraic sets, including the empty boundaries Definition
• Integral elements over a commutative ring and algebraic integers Definition
• Monomials, coefficients, degree in each variable and total degree in F[x₁,…,xₙ] Definition
• Multiplicity of a plane curve at a point Definition
• Plane projective curves and their components Definition
• Polynomials that split and splitting fields of a polynomial or a family of polynomials Definition
• Repeated roots in extension fields and separable polynomials Definition
• Tangent cone and tangent lines at a point Definition
• The group μₙ(K) of n-th roots of unity in a field, and primitive n-th roots of unity Definition
• Normalization of a nodal affine plane curve Example
• Normalizing the cuspidal plane curve Example
• Normalizing the nodal plane curve Example
• The divisor-sum identity at q=2, n=3 finds exactly two monic irreducible cubics Example
• Two transverse cubics meet in nine points Example
• Vanishing sets and vanishing ideals form a contravariant Galois connection Example
• FALSE: if deg f=∑ᵢtᵢ and | Sᵢ|>tᵢ then f is nonzero somewhere on S₁×⋯× Sₙ False statement
• A field isomorphism transports polynomials coefficientwise and carries roots, factorizations, and splitting to roots, factorizations, and splitting Lemma
• A maximal algebraically independent set is a transcendence basis Lemma
• A polynomial vanishing at every tuple from an infinite subdomain is the zero polynomial Lemma
• A vanishing ideal is always radical Lemma
• An ideal and its radical have the same zero locus Lemma
• Chebyshev bounds and endpoint derivatives used by stabilized Runge–Kutta methods Lemma
• Classical Poincare products for A, B, D and I2(m) from the permutation and signed-permutation models Lemma
• Coprime tangent cones force a power of the maximal ideal into the local ideal Lemma
• Curves without a common component meet finitely often Lemma
• Evaluation at a point has kernel (x₁-a₁,..., xₙ-aₙ) Lemma
• f̃ is multilinear, agrees with f at every point of {0,1}ⁿ, is degree-nonincreasing when nonzero, and is the unique multilinear polynomial with that agreement Lemma
• Finite purely inseparable rational extensions admit a finite Frobenius envelope Lemma

…and 22 more results.

Dependency tree · two levels

8 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

• Neil Donaldson, Math 120B Notes, Section 22.4 (standard reference, not scraped)
• James McKernan, MIT 18.703 Lecture 21, Definition 21.4 (standard reference, not scraped)