Polynomial

c_{0} + c_{1} a + ⋯ + c_{n} a^{n} or c_{i} a^{i} .

Homogeneous Polynomial

Degree

Division Algorithm

\begin{matrix} f & ( & x & ) & , g & ( & x & ) & , q & ( & x & ) & , r & ( & x & ) & : & P_{R} & , \\ f & ( & x & ) & = g & ( & x & ) & q & ( & x & ) & + r & ( & x & ) & and  degree & ( & r & ) & < degree & ( & g & ) & . \end{matrix}

q is called the quotient and r is called the remainder.

Example

a^{4} + a + 1 = (a^{2} − 1) (a^{2} + 1) + a + 2.

Multivariate

\begin{matrix} d = & ( & d_{1} & , ⋯, & d_{n} & ) & : & ℕ^{n} & . \\ a^{d} & = & {a_{1}}^{d_{1}} & ⋯ & {a_{n}}^{d_{n}} \end{matrix}

Ordering

\begin{matrix} d, e : & ℕ^{n} & , \\ ( & a^{d} & <_P & a^{e} & ) & = & ( & d < e & ) & . \end{matrix}

\begin{matrix} ( & d < e & ) & + & ( & d = e & ) & + & ( & e < d & ) & ( & total ordering & ) & , \\ ( & f : & ℕ^{n} & ) & d < e → d + f < e + f & ( & well-ordering & ) & . \end{matrix}

Lexicographic Order

(1, 1) < (2, 0), x_{1} x_{2} < {x_{1}}^{2}

Graded Lexicographic Order

\begin{matrix} |d| = & \underset{i}{Σ} & d_{i} & , \\ ( & |d| < |e| & ) & + & ( & ( & |d| = |e| & ) & × & ( & d < e & ) & ) & = & ( & d <_glex e & ) & . \end{matrix}

Example

leading-term_lex(x² + y³) = x². leading-term_glex(x² + y³) = y³.

Ideal

\begin{matrix} I ⊆ R,  r : R,  a : I, \\ r a : I,  a r : I. \end{matrix}

Lemma

\begin{matrix} I ⊆ R,   a, b : I, \\ a − b : I. \end{matrix}

Lemma

\begin{matrix} R : Ring, & f_{i} & : R, & a_{i} & : R, \\ \underset{i}{Σ} & a_{i} & f_{i} & . \end{matrix}

Example

Noetherian Ring

Π I = <f>.

Hilbert’s Basis Theorem

() ()

Example

F = <1>.

Gröbner Basis

\begin{matrix} LT ≡ leading-term,   I ⊆ k[x], & Σ_{f : I} & LT & ( & f & ) & = c & x_{α} & ,   G ⊆ I, \\ <LT & ( & g & ) & > = <LT & ( & I & ) & >. \end{matrix}

Least Common Multiple

LM ≡ leading-monomial LM(f) = Π_i x_i^α_i, LM(g) = Π_i x_i^β_i, γ_i = max(α_i, β_i), x^γ.

S-Polynomial

S (f, g) = x^{γ} f / LT (f) − x^{γ} g / LT (g) .

Example

f = x y - 1, g = y² - 1, LM(f) = x y, LM(g) = y², x^γ = 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

f = g a + r.

Example

Affine Variety

V (f) ≡ (a : F^{n}) → \underset{i}{Π} f_{i} (a) = 0.

V (I) ≡ (a : F^{n}) → (f : I) → f (a) = 0.

Theorem

The Weak Zero Theorem

Lemma

\begin{matrix} V ⊆ & F^{n} & , \\ I & ( & V & ) & ≡ & ( & a : V & ) & → f & ( & a & ) & = 0. \end{matrix}

Hilbert’S Zero Theorem

g^{m} : <f>.

Radical

I^{1 / 2} .

The Strong Zero Theorem

I (V (I)) = I^{1 / 2} .

Elimination Ideal

Factorisation

a^{2} − b^{2} = (a + b) (a − b) .

a^{2} + 2 a b + b^{2} − c^{2} + 2 c d − d^{2} = (a^{2} + 2 a b + b^{2}) − (c^{2} − 2 c d + d^{2}) = {(a + b)}^{2} − {(c − d)}^{2} = (a + b + c − d) (a + b − c + d) .

a^{3} + b^{3} = (a + b) (a^{2} − a b + b^{2}) .

a^{3} − b^{3} = (a − b) (a^{2} + a b + b^{2}) .

a^{4} − b^{4} = (a^{2} + b^{2}) (a^{2} − b^{2}) = (a^{2} + b^{2}) (a + b) (a − b) .

a^{2 n} − b^{2 n} = (a^{n} + b^{n}) (a^{n} − b^{n}) .

a^{n} + b^{n} = (a + b) (a^{n − 1} − a^{n − 2} b)

aⁿ + bⁿ = (a + b) (a^{n - 1} - a^{n - 2} b + a^{n - 3} b² - ⋯ - a b^{n - 2} + b^{n - 1})

Irreducible Polynomial

Uniqueness

Polynomial Decomposition

Polynomial Ring

Formal Derivative

c₀ + c₁ a + c₂ a² ⋯ + c_n aⁿc₁ + 2 c₂ a + ⋯ + n c_n a^{n - 1}.

Polynomial Transformation

Quadratic Form

c_{i, j} a_{i} a_{j} .

Polynomial

Homogeneous Polynomial

Degree

Division Algorithm

Example

Multivariate

Ordering

Lexicographic Order

Graded Lexicographic Order

Example

Ideal

Lemma

Lemma

Example

Noetherian Ring

Hilbert’s Basis Theorem

Example

Gröbner Basis

Least Common Multiple

S-Polynomial

Example

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

Irreducible Polynomial

Uniqueness

Polynomial Decomposition

Polynomial Ring

Formal Derivative

Polynomial Transformation

Quadratic Form

Orthogonal Polynomials

Hermite Polynomials

Jacobi Polynomials

Gegenbauer Polynomials

Chebyshev Polynomials

Legendre Polynomials

Zernike Polynomials

Laguerre Polynomials

References