# Polynomial

## Contents

$c0 + c1 a + ⋯ + cn an or ci ai.$

## Division Algorithm

$f(x), g(x), q(x), r(x) : PR,f(x) = g(x) q(x) + r(x) and degree(r) < degree(g).$

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

#### Example

$a4 + a + 1 = (a2 − 1) (a2 + 1) + a + 2.$

## Multivariate

$d = (d1, ⋯, dn) : ℕn.ad = a1d1 ⋯ andn$

## Ordering

$d, e : ℕn,(ad <_P ae) = (d < e).$$(d < e) + (d = e) + (e < d) (total ordering),(f : ℕn) d < e → d + f < e + f (well-ordering).$

### Lexicographic Order

$(1, 1) < (2, 0), x1 x2 < x12$

$|d| = Σi di,(|d| < |e|) + ((|d| = |e|) × (d < e)) = (d <_glex e).$



## Ideal

$I ⊆ R, r : R, a : I,r a : I, a r : I.$

### Lemma

$I ⊆ R, a, b : I,a − b : I.$

### Lemma

$R : Ring, fi : R, ai : R,Σi ai fi.$

1.

## Noetherian Ring

$Π I = .$

### Hilbert’s Basis Theorem

$is-noetherian-ring(R) → is-noetherian-ring(R[x]).$

### Example

$F = <1>.$

## Gröbner Basis

$LT ≡ leading-term, I ⊆ k[x], Σf : I LT(f) = c xα, G ⊆ I, = .$

## Least Common Multiple

LM ≡ leading-monomial LM(f) = Πi xiαi, LM(g) = Πi xiβi, γi = max(αi, βi), xγ.

## S-Polynomial

$S(f, g) = xγ f / LT(f) − xγ g / LT(g).$

### 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.

## Reduced Gröbner Basis

$leading-coefficient() = 1, p : G, q : G - (p).$

### Lemma

$f = g a + r.$

## Affine Variety

$V(f) ≡ (a : Fn) → Πi fi(a) = 0.$$V(I) ≡ (a : Fn) → (f : I) → f(a) = 0.$

### Lemma

$V ⊆ Fn,I(V) ≡ (a : V) → f(a) = 0.$

### Hilbert’S Zero Theorem

$gm : .$

$I1 / 2.$

### The Strong Zero Theorem

$I(V(I)) = I1 / 2.$

## Factorisation

$a2 − b2 = (a + b) (a − b).$$a2 + 2 a b + b2 − c2 + 2 c d − d2 = (a2 + 2 a b + b2) − (c2 − 2 c d + d2) = (a + b)2 − (c − d)2 = (a + b + c − d) (a + b − c + d).$$a3 + b3 = (a + b) (a2 − a b + b2).$$a3 − b3 = (a − b) (a2 + a b + b2).$$a4 − b4 = (a2 + b2) (a2 − b2) = (a2 + b2) (a + b) (a − b).$$a2 n − b2 n = (an + bn) (an − bn).$$an + bn = (a + b) (an − 1 − an − 2 b)$

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

## Formal Derivative

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

## Polynomial Transformation

$ci, j ai aj.$