The euclidean algorithm and bézouts theorem

HANOI PEDAGOGICAL UNIVERSITY 2DEPARTMENT OF MATHEMATICS LE THI TRANG THE EUCLIDEAN ALGORITHM BACHELOR THESIS Hanoi – 2019... HANOI PEDAGOGICAL UNIVERSITY 2DEPARTMENT OF MATHEMATICS LE TH

HANOI PEDAGOGICAL UNIVERSITY 2

DEPARTMENT OF MATHEMATICS

LE THI TRANG

THE EUCLIDEAN ALGORITHM

BACHELOR THESIS

Hanoi – 2019

HANOI PEDAGOGICAL UNIVERSITY 2

DEPARTMENT OF MATHEMATICS

LE THI TRANG

THE EUCLIDEAN ALGORITHM

BACHELOR THESIS

SUPERVISOR:

Dr NGUYEN TAT THANG

Hanoi – 2019

I here by declare that the data and the results of this thesis are true and not identical to other theses I also assure that all the help for this thesis has been acknowledged and that the results presented in the thesis has been identified clearly.

Ha noi, May 5, 2019

Student

Le Thi Trang

First and foremost, my heartfelt goes to my admirable supervisor, Mr Nguyen Tat Thang (Institute of Mathematics, Vietnam Academy of Science and Technol- ogy), for his continuous supports when I met obstacles during the journey The completion of this study would not have been possible without his expert advice, close attention and unswerving guidance.

Secondly, I am keen to express my deep gratitude for my family for encourage

me to continue this thesis I owe my special thanks to my parents for their emotional and material sacrifices as well as their understanding and unconditional support Finally, I would like to take this opportunity to thank to all teachers of the Department of Mathematics, Hanoi Pedagogical University No.2, the teachers in the geometry group as well as the teachers involved The lecturers have imparted valuable knowledge and facilitated for me to complete the course and the thesis I

am specially thankful to all my best friends at university for endless encouragement Due to time, capacity and conditions are limited, so the thesis cannot avoid errors Therefore, I look forward to receiving valuable comments and recommenda- tions from teachers and friends.

Ha noi, May 5, 2019

Student

Le Thi Trang

Preface 1

1.1 Affine algebraic curves 3

1.2 Projective plane algebraic curves in P2 61.3 The Euclid’s Algorithm 8

Bibliography 42

Algebraic geometry is the study of zero sets of polynomials One of

the important theorems in algebraic geometry is the B´ezout’s

Theo-rem (TheoTheo-rem 2.3.2), which explains the intersections of two algebraic

plane curves in the (projective) plane

In circa the year 300 BC Euclid of Alexandria (∼ 325– ∼ 265 BC)

wrote the treatise The Elements consisting of thirteen books Book

seven is an introduction to number theory and it contains the

Eu-clidean algorithm to find the greatest common divisor of two integers

This algorithm is one of the oldest in history and is still in common

use

In the year 1748 Leonhard Euler (1707–1783) and Gabriel Cramer

(1704–1752) already stated B´ezout’s Theorem, but neither of them

succeeded in completing a proof A few years later, in the year 1764,

´

Etienne B´ezout (1730–1783) gave the first satisfactory proof as a

re-sult of earlier work of Colin Maclaurin (1698–1746) In actual fact,

this proof was incomplete in the count of multiple points The proper

count of multiplicities was settled more then one hundred years later,

in the year 1873, by Georges-Henri Halphen (1844–1889) This

his-torical information can be found in two books: The first is ”The

Mac-Tutor History of Mathematics” of John J O’Connor and Edmund F

Robertson ; And the second is ”A history of Mathematics” of Uta C

Merzbach and Carl B Boyer

Two years ago, in 2009, Jan Hilmar and Chris Smyth finished their

article [1] and in this work they proved B´ezout’s theorem using the

Euclidean algorithm This bachelor thesis is based on the article [1]

The aim of this bachelor thesis is to represent a proof of the B´ezout’s

theorem using the Euclidean algorithm The thesis consists of three

chapters:

In the first chapter, we will recall the definition of algebraic curves

We also represent the Euclidean algorithm for numbers

Chapter 2 consists of definitions of the matrix Sylvester, the

resul-tant of polynomials, intersection multiplicity and some imporresul-tant its

properties With these homogeneous polynomials, we will formulate

the B´ezout’s theorem and represent its proof in terms of resultants

In chapter 3, we provide another proof of B´ezout’s theorem which

we use the idea of Euclidean algorithm For this, we need to construct

an Euclidean algorithm for curves This chapter is base on the article

[1]

Suppose that f (x, y) is a two-variable polynomial, different from a

con-stant, with complex coefficients We say that f (x, y) has no multiple

factor if there does not exist an expansion:

Hilbert’s Nullstellensatz theorem:

Let f (x, y) and g(x, y) be polynomials with complex coefficients, we

have

(x, y) ∈ C2 | f (x, y) = 0 = (x, y) ∈ C2 | g(x, y) = 0

if and only if there exist positive integers m, n such that f divides gn(gn is divisible by f ) and g divides fm.

The Hilbert’s Nullstellensatz theorem: Let K = K be an algebraicallyclosed field,

two variables Hence, two polynomials are equivalent if and only if

each is a scalar multiple of the other

Example 1.4 Consider the following polynomials:

f (x, y) = x4 + 4x3y2 + 4x2y4 = x2(x + 2y2)2,g(x, y) = x4 + 2x3y2 = x3(x + 2y2)

We see that f2 is divisible by g and g2 is divisible by f Therefore, fand g define the same affine algebraic curve:

(x, y) ∈ C2 | f (x, y) = 0 = (x, y) ∈ C2 | g(x, y) = 0

Definition 1.5 Let f (x, y) be a polynomial in two variables

f (x, y) = P

i,j

ai,jxiyj

The degree d of the curve C = (x, y) ∈ C2 | f (x, y) = 0 is the degree

of polynomial f (x, y), it means that

d = max {i + j | ai,j 6= 0}

Definition 1.6 A curve is defined by a linear equation:

ax + by + c = 0

where a, b, c are complex numbers, (a2 + b2) 6= 0, is called by a line

Definition 1.7 A non-constant polynomial in n variables

f (x1, x2, , xn) is called homogeneous of degree d if for all λ ∈ Cthen f (λx1, λx2, , λxn) = λnf (x1, x2, , xn)

Lemma 1.8 f is homogeneous of degree d if and only if f can be

written in the following form:

Proposition 1.10 Suppose that f (x, y) is a non-constant polynomial

in two variables, homogeneous of degree d with complex coefficients

Then it can be factored into a product of linear polynomials

f (x, y) =

nQi=1(αix − βiy),

for α, β ∈ C

Proof Since f (x, y) is homogeneous of degree d, so:

f (x, y) =

nPi=0

ai xi yd−i = yd

dPi=0

ai (xy)i,

where a21+ a22 + + a2n 6= 0 Suppose that e is the largest number in{0, 1, , d} such that ae 6= 0, we obtain,

dPi=0

ai(xy)i

is a e-degree polynomial in one variable with complex coefficients

Then, it will factor as,

dPi=0

ai(xy)i = ae

eQi=1(xy − λi),

foll all λi ∈ C Therefore:

f (x, y) = ae yd

eQi=1(xy − λi) = ae yd−e

eQi=1(x − λiy)

Hence, the proof is complete

A curve in C2 is never compact, but we can compactify it by adding

in the ”points at infinity” and getting a projective space

For example consider two parallel lines which lie in C2 They will ofcourse never intersect but if we add a point at infinity to each line

they meet at a distinct point

Definition 1.11 (Complex projective spaces) A n-dimensional

com-plex projective space Pn is set of complex subspaces of dimension 1 of

a vector space Cn+1

When n = 1 we get the complex projective line and when n = 2 we

get the complex projective plane as above

Remark 1.12 If V is a vector space over any field K then the

cor-responding projective space P(V ) is the set of vector subspaces ofdimension 1 of V

In above definition, K = C, V = Cn+1 and for simplicity, we oftenwrite Pn instead of P(Cn+1)

Remark 1.13 The projective plane P2 is the set of complex subspaces

of dimension 1 of C3:

P2 = [x, y, z]|(x, y, z) ∈ C3\ {0} [x, y, z] = [u, v, w] if and only if there exists λ ∈ C\ {0} such that

x = λu, y = λv, z = λw

Definition 1.14 Let F (x, y, z) be a non-constant homogeneous

poly-nomial with complex coefficients Suppose that F (x, y, z) has no

mul-tiple factor Then a projective plane algebraic curve C defined by

F (x, y, z) is C = [x, y, z] ∈ P2|F (x, y, z) = 0

(We call C is the projective closure of C)

Definition 1.15 The projective plane algebraic curve corresponding

to an affine algebraic curve C over a field K is called the projective

closure of C in P2(K)

Example 1.16 For K = R

C : x4 − 5x2y2 + 4y4 + 5x2 + 9y + 10 = 0

C : X4 − 5X2Y2 + 4Y4 + 5X2Z2 + 9Y Z3 + 10Z4 = 0

Remark 1.17 Since F is a homogeneous polynomial, so for all λ ∈

C\ {0} then

F (λx, λy, λz) = 0 ←→ F (x, y, z) = 0

Therefore, the condition F (x, y, z) = 0 does not depend on the choice

of homogeneous coordinates (x, y, z)

Remark 1.18 Similarly in C2, two homogeneous polynomials with

no common factor F (x, y, z) and G(x, y, z) define the same projective

curve in P2 if and only if each is a scalar multiple of the other

A homogeneous polynomial with multiple factors can be viewed as a

curve with multiple components

Definition 1.19 The degree of a projective plane algebraic curve C

in P2 defining by the homogeneous polynomial F (x, y, z) is the degree

of F (x, y, z)

Definition 1.20 A curve C is called irreducible if F (x, y, z) is

irre-ducible

A irreducible polynomial F (x, y, z) is a non-constant polynomial that

cannot be factored into the product of two non-constant polynomials

i.e, we say that F (x, y, z) is irreducible if the only divisors of F (x, y, z)

are a constant and itself

The Euclidean algorithm, also known as Euclid’s algorithm, is an

al-gorithm for finding the greatest common divisor (GCD) between two

integer numbers The GCD is the largest number that divides two

numbers without a remainder The GCD of two numbers can be

found by making a list of factors for the two numbers, and finding

the largest factor that is in both sets This works well for small

num-bers, but it can become quite tedious and time consuming for larger

numbers To address this problem, Euclid’s algorithm can be used,

which allows for the GCD of large numbers to be found much faster

Euclid’s algorithm uses the principle that the GCD of a set of two

numbers does not change if you replace the larger of the two with the

remainder when you divide the larger of two by the smaller

Now, we present the Euclidean algorithm: The Euclidean algorithm

proceeds in a series of steps such that the output of each step is used

as an input for the next one Let k be an integer that counts the steps

of the algorithm, starting with zero Thus, the initial step corresponds

to k = 0, the next step corresponds to k = 1, and so on

Each step begins with two non-negative remainders rk−1 and rk−2.Since the algorithm ensures that the remainders decrease steadily with

every step, rk−1 is less than its predecessor rk−2 The goal of the k-thstep is to find a quotient qk and remainder rk that satisfy the equation

rk−2 = qkrk−1 + rk

and that have rk < rk−1 In other words, multiples of the smallernumber rk−1 are subtracted from the larger number rk−2 until theremainder rk is smaller than rk−1

In the initial step (k = 0), the remainders r−2 and r−1 equal a and b(suppose that a > b), the numbers for which the GCD is sought In

the next step (k = 1), the remainders equal b and the remainder r0 ofthe initial step, and so on Thus, the algorithm can be written as a

sequence of equation

Step 0 : a = q0b + r0Step 1 : b = q1r0 + r1Step 2 : r0 = q2r1 + r2Step 3 : r1 = q3r2 + r3

Step n : rn−2 = qnrn−1+ 0

The algorithm stops when a zero remainder rn = 0 appears for thefirst time The last non-zero remainder is rn−1 We check that rn−1 isthe greatest common divisor From the identity

Hence the last non-zero remainder rn−1 is the greatest common divisor.

Example 1.21 Let a = 2322, b = 654 We have:

B´ ezout’s Theorem

In this chapter, we formulate the B´ezout’s theorem which provide us

the number of intersection points of two algebraic curves We also

give the proof of the theorem in terms of the resultant

Definition 2.1 Let two polynomials F, G ∈ C[x]:

F (x) = a0xm+ a1xm−1 + + am for a0 6= 0,

G(x) = b0xn+ b1xn−1+ + bn for b0 6= 0

The matrix Sylvester (Syl) of F and G with the variable x is the (m +

n) × (m + n) matrix given by:

a0 a1 am

b0 b1 bn

b0 b1 bn

for the vacant positions in matrix equals 0

The first n rows of the matrix consist of shifts of (a0, a1, , am), andthe last m rows of it consist of shifts of (b0, b1, , bn)

Then the resultant of F and G is defined as the determinant of the

matrix Sylvester

Res(F,G,x) = det Syl(F,G,x)

Let F (x, y, z) =

mPi=1

ai(y, z)xi, Q(x, y, z) =

nPj=1

aj(y, z)xj ∈ C[x, y, x] =(C[y, z])(x) Then, the resultant Res(F,G,y,z) ∈ C[y, z] of F and G isdefined similarly

Example 2.2 a) Let F (x) = ax2+ bx + c and G(x) = 2ax + b Then

Res(F, G, x) =

2a b 0

0 2a b

−3 y y22y y2 0

0 2y y2

= −y4

Proposition 2.3 Let F (x), G(x) ∈ C[x], then F (x) and G(x) have

a nonconstant common factor if and only if Res(F, G, x) = 0

Proof Suppose that

F (x) = a0xm+ a1xm−1 + + am for a0 6= 0,

G(x) = b0xn+ b1xn−1+ + bn for b0 6= 0

are the x-variable polynomials with degree m, n, respectively Then

F (x) and G(x) have a nonconstant common factor R(x) if and only if

there exists the polynomials φ(x) and ψ(x) such that:

F (x) = R(x)φ(x), G(x) = R(x)ψ(x),

where φ(x), ψ(x) are not identically zero and 0 ≤ deg φ(x) ≤ m, 0 ≤

deg ψ(x) ≤ n:

φ(x) = α0xm−1 + α1xm−2 + + αm−1ψ(x) = β0xn−1 + β1xn−2 + + βn−1

It happen if and only if

It is easy to see that the system (2.1.1) is equivalent to the

homoge-neous linear equation system having a non-zero solution It is

η = (β0, β1, , βn−1, −α0, −α1, , −αm−1)

That happens if and only if

det Sylt(F, G, x)
= 0

Hence, Res(F, G, x) = det Syl(F, G, x)
= det Sylt(F, G, x)
= 0

Example 2.4 a) Let F, G be polynomials in one variable and

F (x) = (x + 1)(x + 2) = x2 + 3x + 2G(x) = x + 1

F and G have a common factor that is x + 1, then

Res(F, G, x) =

...

are the x-variable polynomials with degree m, n, respectively Then

F (x) and G(x) have a nonconstant common factor R(x) if and only if

there exists the polynomials φ(x) and ψ(x)...

= −y4

Proposition 2.3 Let F (x), G(x) ∈ C[x], then F (x) and G(x) have

a nonconstant common factor if and only if Res(F, G, x) =

Proof Suppose that

F (x)... happen if and only if

It is easy to see that the system (2.1.1) is equivalent to the

homoge-neous