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
• R[x] is Noetherian if and only if R is Noetherian Corollary
• 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
• 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
• Polynomials that split and splitting fields of a polynomial or a family of polynomials Definition
• Repeated roots in extension fields and separable polynomials Definition
• The group μₙ(K) of n-th roots of unity in a field, and primitive n-th roots of unity Definition
• The divisor-sum identity at q=2, n=3 finds exactly two monic irreducible cubics 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
• 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
• For a finite group of ring automorphisms the orbit polynomial is monic over the invariant subring, so the ring is integral over its invariants Lemma
• For prime q and d≥1, the congruence xᵈ≡1 (mod q) has at most d residue-class solutions Lemma
• If F does not shatter T then x_T agrees on {v_F:F inF} with a combination of the x_S for S⊊ T Lemma
• Reducing f modulo gᵢ(xᵢ)=∏_s∈ Sᵢ(xᵢ-s) lowers each deg_xᵢ below | Sᵢ|, preserves the values on the grid, and preserves any top-degree coefficient whose exponents stay below the grid sizes Lemma
• The Rabinowitsch auxiliary ideal has no common zero Lemma
• Φ₁(0)=-1 and Φₙ(0)=1 for n≥2 Lemma
• Φ_pʳ(t)=∑_k<pt^kpʳ⁻¹, and Φ_pʳ(t+1) is Eisenstein at p Proposition
• Formal polynomials are not the functions they induce Remark
• Alon's Combinatorial Nullstellensatz: if deg f=∑ᵢtᵢ, the coefficient of x₁^t₁⋯ xₙ^tₙ in f is nonzero, and | Sᵢ|>tᵢ, then f(s₁,…,sₙ)≠0 for some sᵢ∈ Sᵢ Theorem
• Covering {0,1}ⁿ minus the origin by affine hyperplanes avoiding the origin needs at least n of them Theorem
• For every finite-dimensional space, σ_F(T) is exactly the set of roots in F of χ_T Theorem
• For every n≥1 there are infinitely many primes p with p≡1 (mod n) Theorem
• For monic f, Res(f,g)=∏ᵢ g(αᵢ) and it vanishes exactly when f and g have a common root Theorem
• If deg_xᵢP<| Sᵢ| for each i and P vanishes on S₁×⋯× Sₙ, then P=0 Theorem
• Rational root theorem Theorem
• The Schwartz-Zippel lemma Theorem

…and 1 more result.

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)