Polynomial

Contents
Homogeneous Polynomial

Degree

Division Algorithm
q is called the quotient and r is called the remainder.
Example
Multivariate
Ordering
Lexicographic Order
Graded Lexicographic Order
Example
leading-termlex(x2 + y3) = x2. leading-termglex(x2 + y3) = y3.
Ideal
Lemma
Lemma
Example
1.
Noetherian Ring
Hilbert’s Basis Theorem
Example
Gröbner Basis

Least Common Multiple
LM ≡ leading-monomial LM(f) = Πi xiαi, LM(g) = Πi xiβi, γi = max(αi, βi), xγ.
S-Polynomial
Example
f = x y - 1, g = y2 - 1, LM(f) = x y, LM(g) = y2, xγ = x y2, S(f, g) = x y2 f / x y - x y2 g / y2= y (x y - 1) - x (y2 - 1) = x - y.
Buchberger’s Algorithm
Reduced Gröbner Basis
Example
Lemma
Example
Affine Variety
Theorem
The Weak Zero Theorem
Lemma
Lemma
Hilbert’S Zero Theorem
Radical
The Strong Zero Theorem
Elimination Ideal
Factorisation

an + bn = (a + b) (an - 1 - an - 2 b + an - 3 b2 - ⋯ - a bn - 2 + bn - 1)
Irreducible Polynomial

Uniqueness

Polynomial Decomposition

Polynomial Ring


c0 + c1 a + c2 a2 ⋯ + cn anc1 + 2 c2 a + ⋯ + n cn an - 1.


Orthogonal Polynomials
Hermite Polynomials
Jacobi Polynomials
Gegenbauer Polynomials
Chebyshev Polynomials
Legendre Polynomials
Zernike Polynomials
Laguerre Polynomials
References