The euclidean algorithm and bézouts theorem

MINISTRY OF EDUCATION AND TRAINING HANOI PEDAGOGICAL UNIVERSITY DEPARTMENT OF MATHEMATICS LE THI TRANG THE EUCLIDEAN ALGORITHM ´ AND BEZOUT’S THEOREM BACHELOR THESIS Hanoi – 2019 MINISTRY OF EDUCATION AND TRAINING HANOI PEDAGOGICAL UNIVERSITY DEPARTMENT OF MATHEMATICS LE THI TRANG THE EUCLIDEAN ALGORITHM ´ AND BEZOUT’S THEOREM BACHELOR THESIS SUPERVISOR: Dr NGUYEN TAT THANG Hanoi – 2019 Declarations 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 Acknowledgements First and foremost, my heartfelt goes to my admirable supervisor, Mr Nguyen Tat Thang (Institute of Mathematics, Vietnam Academy of Science and Technology), 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 recommendations from teachers and friends Ha noi, May 5, 2019 Student Le Thi Trang Contents Preface 1 Priliminaries 1.1 Affine algebraic curves 1.2 Projective plane algebraic curves in P2 1.3 The Euclid’s Algorithm B´ ezout’s Theorem 12 2.1 Resultant of polynomials 12 2.2 Intersection multiplicity 20 2.3 B´ezout’s Theorem 22 Euclidean Algorithm and B´ ezout Theorem 3.1 Intersection cycles 3.2 An intersection algorithm base on the Euclidean Algo- 24 24 rithm 27 3.3 Some examples 32 3.4 Proof of B´ezout Theorem 39 Conclusions 41 CONTENTS Bibliography 42 Bachelor thesis le thi trang Preface Algebraic geometry is the study of zero sets of polynomials One of the important theorems in algebraic geometry is the B´ezout’s Theorem (Theorem 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 Euclidean 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 result 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 historical information can be found in two books: The first is ”The MacTutor 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 Bachelor thesis le thi trang 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 consists of definitions of the matrix Sylvester, the resultant of polynomials, intersection multiplicity and some important 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] Chapter Priliminaries 1.1 Affine algebraic curves Suppose that f (x, y) is a two-variable polynomial, different from a constant, with complex coefficients We say that f (x, y) has no multiple factor if there does not exist an expansion: f (x, y) = g (x, y)h(x, y) Definition 1.1 Let f (x, y) ∈ C[x, y] be a non-constant polynomial with no multiple factor Then an affine algebraic curve C in C2 defined by f (x, y) is C = (x, y) ∈ C2 | f (x, y) = Remark 1.2 In definition 1.1.1, f (x, y) has no multiple factor by 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) = = (x, y) ∈ C2 | g(x, y) = Bachelor thesis le thi trang if and only if there exist positive integers m, n such that f divides g n (g n is divisible by f ) and g divides f m The Hilbert’s Nullstellensatz theorem: Let K = K be an algebraically closed field, I = f = {h.f | h ∈ K[x1 , x2 , , xn ]} V (I) = {x ∈ Kn | f (x) = ∀f ∈ I} and let √ I = {f ∈ K[x1 , x2 , , xn ] : f m ∈ I for some m ≥ 1}: The radical ideal of I Then √ I V (I) = I Remark 1.3 In general, to define an affine algebraic curve in C2 is equivalent to give a equivalence class of non-constant polynomials in 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 + 4x3 y + 4x2 y = x2 (x + 2y )2 , g(x, y) = x4 + 2x3 y = x3 (x + 2y ) We see that f is divisible by g and g is divisible by f Therefore, f and g define the same affine algebraic curve: (x, y) ∈ C2 | f (x, y) = = (x, y) ∈ C2 | g(x, y) = Definition 1.5 Let f (x, y) be a polynomial in two variables Bachelor thesis le thi trang and ≤ ∂x r ≤ ∂x G and q, r = Since the coefficients of q and r are the rational functions of y and z, then we can multiply through by the least common multiple H ∈ C[y, z] of their denominators to get: HF = QG + R where Q = qH, R = rH ∈ C[x, y, z] are both homogeneous Since HF is homogeneous, too, so ∂(QG) = ∂R Next, suppose that I = gcd(G, R) As gcd(F, G) = 1, it is implies that gcd(G, H) = Therefore, we can divide though I to obtain H F = QG + R (3) where now G = G I, R = R I, H = H I and gcd(G , R )=gcd(G , H ) = Taking now the intersection cycle of both sides of this with G and using Proposition 3.1.3, we get G (H F ) = G (QG + R ) G H + G F = G R (by Proposition 3.2.b-c) G F = G R − G H Since G = G I, we see that then F G = F (G I) = G R − G H + F I (4) Note that the right-hand side of (4) consists of intersection cycles of curves of lower x-degree and intersection cycles of curves of the form 28 Bachelor thesis le thi trang A.B with ∂x B = 0, as ∂x I = Setting: C−1 = F, C0 = G, D0 = H , C1 = R In the following, as above, a indicates that the gcd has been removed from the polynomials Repeatedly applying (3.2.1.1), we get D0 Ca− = Q0 C0 + C1 D1 C0 = Q1 C1 + C2 Dn−1 Cn−2 = Qn−1 Cn−1 + Cn where ∂x Cn−1 ≥ and ∂x Cn ≥ The corresponding expressions for the curves will read C−1 C0 = C0 C1 − C0 D0 + C−1 I0 C0 C1 = C1 C2 − C1 D1 + C0 I1 ACn−2 Cn−1 = Cn−1 Cn − Cn−1 Dn−1 + Cn−2 In Collecting these expressions together, we obtain n−1 (Ci−1 Ii − Ci Di ) − C0 D0 F G = Cn−1 Cn + C−1 I0 + i=1 As all Ii , Di ∈ C[y, z] and ∂Ii = 0, ∂Di = 0∀i = 0, n − We have thus reduced the problem of computing F G to the problem of computing A.B, where ∂x B = Lemma 3.3 Let F, G be two algebraic curves The intersection cycle 29 Bachelor thesis le thi trang F G is a sum or difference of simpler sums of the following types: (1) The point (1,0,0); (2) A sum (α, 1, 0) , the sum being taken over roots α of a monic α polynomial f ∈ C[x] irreducible over C; let us denote this sum by C0 f (x) ; (3) A double sum (γ, β, 1), where β γ β is taken over the roots β of some monic polynomial g ∈ C[y] irreducible over C, and where γ is taken over the roots γ of some monic polynomial hβ ∈ C(β)[x] irreducible over C(β) In this case, we can write hβ as a 2-variable polynomial h(x, β) with coefficients in C, where the β-degree of h is less than the degree of g; denote our double sum by C1 (h(x, y), g(y) Thus h and g will specify this intersection cycle canonically Proof Let A ∈ C[x, y, z] and B ∈ C[y, z] Firstly, we can assume that B is irreducible over C( by Proposition 3.2.b) If B doesn’t contain the variable y, then, being irreducible, B = z Otherwise, over C, we may writeit as (y − βz) B(y, z) = β where, β are solutions in C of B(y, 1) Moreover, we have the expansion of A(x, y, z) in two different ways: A(x, y, z) = A(x, y, 0) + zA (x, y, z) and A(x, y, z) = A(x, βz, z) + (y − βz)A (x, y, z) 30 Bachelor thesis le thi trang where A , A ∈ C[x, y, z] Using Proposition 3.2.c, we get A.z = A(x, y, 0) + zA (x, y, z) z = A(x, y, 0).z and A.(y − βz) = A(x, βz, z).(y − βz) i.e, (y − βz) = A β A(x, βz, z).(y − βz) β Hence, we have A.B =    A(x, y, 0).z if B = z; A(x, βz, z).(y − βz) otherwise    β The case 1: B = z By factorizing A(x, y, 0) first into irreducible factors over C, we have A1 B =    (1, 0, 0)    if A1 = y; (x − αy).z = α (α, 1, 0) otherwise α where the α are the roots of A1 (x, 1) (y − βz) The case 2: B = β We first factorize A(z, βz, z) over C(β) Taking A2 (x, z) as a typical 31 Bachelor thesis le thi trang factor, we have that either A2 = z and z.(y − βz) = (∂B)(1, 0, 0) A2 B = β (x − γz), where the γ are the or that, over C, we have A2 (x, z) = γ roots in C of A2 (x, 1), and (x − γz).(y − βz) = A2 B = γ β 3.3 (γ, β, 1) β γ Some examples Example 3.4 Take F (x, y, z) = y z − x3 G(x, y, z) = y z − x2 (x + z) Thus the equations F = and G = are homogenized versions of the cubic curves y = x3 and y = x2 (x + 1), plotted in Figure 32 Bachelor thesis le thi trang Figure 1.The ’slice’ z = of the cubic curves y z − x3 (solid line) and y z − x2 (x + z) (dotted line) near (0, 0, 1), an intersection point of multiplicity 4.(There are the curves y = x3 and y = x2 (x + 1).) We can see that they intersect at the origin (0, 0, 1), but it is not immediately clear what the multiplicity of intersection there is And are there other intersection points? Applying Euclidean algorithm to F and G as polynomials in x, we first have F (x, y, z) = G(x, y, z) + x2 z, so that F G = F (x2 z) = 2(F x) + F z (by Proposition 3.2.c-b) We have F x = (y z − x3 ).x = (y z).x = 2(y.z) + (z.x) = 2(0, 0, 1) + (0, 1, 0) (using Proposition 3.2.d) And F z = (y z − x3 ).z = (x3 ).z = 3(0, 1, 0) Therefore, we get F G = 4(0, 0, 1) + 5(0, 1, 0) Thus F and G intersect at (0, 0, 1) with 33 Bachelor thesis le thi trang multiplicity (see Figure 1) and at (0, 1, 0) with multiplicity (see Figure 2) Since both curves have degree 3, and + = × 3, we have checked out B´ezout’s Theorem for this example We have (0, 0, 1) = C1 (x, y) and (0, 1, 0) = C0 (x) Figure The ’slice’ y = of the same curves y z − x3 (solid line) and y z − x2 (x + z) (dotted line) near (0, 1, 0), an intersection point of multiplicity 5.(There are the curves z = x3 and z = x3 /(1 − x2 ).) Example 3.5 Take A(x, y, z) = yz − x2 B(x, y, z) = yz − x(x − z) Thus the equations A = and B = are homogenized versions of the curves y = x2 and y = x(x + 2) It is easy to see that they intersect at the origin (0, 0, 1), but it is not immediately clear what the multiplicity of intersection there is And are there other intersection points? 34 Bachelor thesis le thi trang Applying Euclidean algorithm to A and B as polynomials in x, we first have B(x, y, z) = A(x, y, z) + xz, so that A.B = A.(xz) = A.x + A.z.(by Proposition 3.2.c-b) We have A.x = (yz − x2 ).x = (yz).x = y.z + z.x = (0, 0, 1) + (0, 1, 0) (using Proposition 3.2.d) And A.z = (yz − x2 ).z = (x2 ).z = 2(0, 1, 0) Therefore, we get A.B = (0, 0, 1) + 3(0, 1, 0) Thus A and B intersect at (0, 0, 1) with multiplicity and at (0, 1, 0) with multiplicity So the intersection number of the curves A and B is + = = × Example 3.6 Take F (x, y, z) = (y − z)x5 + (y − yz)x4 + (y − y z)x3 + (−y z + yz )x2 + (−y z + y z )x − y z + y z G(x, y, z) = (y − 2z )x2 + (y − 2yz )x + y − y z − 2z We have: F (x, y, z) = (y − z)x5 + (y − yz)x4 + (y − y z)x3 + (−y z + yz )x2 + (−y z + y z )x − y z + y z =(y − z)(x5 + yz + y x3 − yz x2 − y z x − y z ) G(x, y, z) = (y − 2z )x2 + (y − 2yz )x + y − y z − 2z = (y − 2z )(x2 + yx + y + z ) 35 Bachelor thesis le thi trang Applying one step of Euclidean Algorithm to F and G as polynomials in x, we get F = (y − z)x(x2 − z ) G + z (y − z)(z x − y ); 2 y − 2z Multiply both sides with H = y − 2z ∈ C[y, z], we obtain (y − 2z )F = (y − z)x(x2 − z )G + z (y − z)(z x − y )(y − 2z ) We can see that I =gcd (y − 2z , G) = y − 2z and then, application (4) give F G = G R − G H + F I = G R + F I (since H = 1) where G (x, y, z) = x2 + xy + y + z , R (x, y, z) = z (y − z)(z x − y ), I(y, z) = y − 2z Repeating the process with G and R , we get G = (y + z )(y + z ) xz + y(y + z ) R + z (y − z) z4 Then, z (y − z)G = xz + y(y + z ) R + z (y + z )(y + z )(y − z) i.e., z G = (z − z y ) xz + y(y + z ) + (y + z )(y + z ) Applying (3), we obtain R G = (x2 + yx + y + z ).z (y − z) − z (−y + xz ) 36 Bachelor thesis le thi trang + (−y + xz ) (y + z )(y + z ) = (x2 + yx + y + z ).z + (x2 + yx + y + z ).(y − z) −12(z.y) + (−y + xz ).(y + z ) + (−y + xz ).(y + z ) (x − αy).z + =2 α:α2 +α+1=0 (x − γy).(y − z) γ:γ +γ+1=0 (−y + xz ).(y − βz) + + β:β +1=0 (−y + xz ).(y − βz) β:β +1=0 −12(1, 0, 0) =2 (γ, 1, 1) − 12(1, 0, 0) (α, 1, 0) + α:α2 +α+1=0 γ:γ +γ+1=0 (−(βz)3 + xz ).(y − βz) + + β:β +1=0 We have (−(βz)3 + xz ).(y − βz) β:β +1=0 (γ, 1, 1) = 2C0 (x2 + x + 1) (α, 1, 0) + α:α2 +α+1=0 γ:γ +γ+1=0 +C1 (x2 + x + 2, y − 1), (−(βz)3 + xz ).(y − βz) = 4(1, 0, 0) + C1 (x + y, y + 1), β:β +1=0 (−(βz)3 + xz ).(y − βz) = 8(1, 0, 0) + C1 (x − y , y + 1) β:β +1=0 Thus R G = 2C0 (x2 + x + 1) + C1 (x2 + x + 2, y − 1) +C1 (x + y, y + 1) + C1 (x − y , y + 1) Similarly, we also compute F I = F (x, y, z).(y − 2z ) = F (x, βz, z).(y − βz) β:β −2=0 37 Bachelor thesis le thi trang = C1 (x3 − y, y − 2) + C1 (x2 + yx + 2, y − 2) + 2(1, 0, 0) Hence, F G = 2(1, 0, 0) + 2C0 (x2 + x + 1) + C1 (x + y, y + 1) +C1 (x − y , y + 1) + C1 (x2 + x + 2, y − 1) + C1 (x3 − y, y − 2) +C1 (x2 + yx + 2, y − 2) Note that the multiplicities add up to 2+4+2+2+4+6+4 = 24 = 6×4 Example 3.7 Let the curves A and B be given by: A(x, y, z) = x3 y − xz + y z B(x, y, z) = xy + xz + yz Using the Euclidean Algorithm for F and G as polynomials in x, we write HF = QG + R with R(y, z) = yz (3y + 4y z + 3z y + z ) H(y, z) = (y + z)3 We have gcd (B, R) = gcd( B, H) = 1, then, A.B = B.R − B.H + A.I = B.R − B.H (since I = 1) We can compute: B.H = (xy + yz + zx).(y + z)3 = 6(1, 0, 0), B.R = B.(yz (3y + 4y z + 3z y + z )) = B.y + B.z + B.(3y + 4y z + 3z y + z ) 38 Bachelor thesis le thi trang We have B(x, y, z).y = (1, 0, 0) + (0, 0, 1) B(x, y, z).z = 4(1, 0, 0) + 4(0, 1, 0) B(x, y, z).(3y +4y z+3z y+z ) = B(x, βz, z).(y−βz) β:3β +4β +3β+1=0 = C1 (x + 3y +4y +3y+1 1+3y +4y +3y+1 , 3y + 4y + 3y + 1) + 3(1, 0, 0) = C1 (x(1 + 3y + 4y + 3y + 1) + 3y + 4y + 3y + 1, 3y + 4y + 3y + 1) + 3(1, 0, 0) = C1 (x, 3y + 4y + 3y + 1) + 3(1, 0, 0) Therefore, we obtain A.B = 2(1, 0, 0) + 4(0, 1, 0) + (0, 0, 1) + C1 (x, 3y + 4y + 3y + 1) Note that the multiplicities add up to + + + = 10 = × 3.4 Proof of B´ ezout Theorem In this section, we use the algorithm describe in Section 3.2 to give a simple proof of B´ezout Theorem (Theorem 2.11) Proof We need show that #(F G) = iP (F, G) = mn We P proceed by induction on the x-degree of G Firstly, suppose that G has x-degree Then over C, G factors as a product of n lines L Therefore,F G is a sum of n intersection cycles F L From Section 3.2, each F L is equal to F L, where F is a polynomial in two variables of degree m, and thus a product of m lines Hence F L can be written as a sum of m intersections L L, giving mn such intersections in total Since, by Proposition 3.1.3.d, the intersection cycle L L is the single point, we have #(F G) = mn Suppose now that G has x39 Bachelor thesis le thi trang degree k > and that we know that the result holds for all G with ∂x G < k and for all F Then, by (3.1.2), #(F G) = #(G R ) − #(G H ) + #(F I) =(∂R − ∂H ) ∂G + ∂F ∂I, note that ∂x R < ∂x G = k and ∂x H = ∂x I = Using the fact that all polynomials involved are homogeneous, from (3), we have ∂R − ∂H = ∂F Moreover, ∂G + ∂I = ∂G (since G = G I) Therefore, we obtain the result #(F G) = ∂F (∂G − ∂I) + ∂F ∂I = ∂F ∂G = mn follows for ∂x G = k 40 Bachelor thesis le thi trang Conclusion In this thesis, we have presented systematically the following results Recall some definitions: affine algebraic curves, homogeneous polynomials, degree of curves, complex projective spaces, projective algebraic curves, We have represented the Euclid’s algorithm for finding the greatest common divisor between two integer numbers We have shown the B´ezout’s theorem which states that the number of common points of two algebraic curves in projective plane is exactly equal to the product of their degrees We construct the Euclidean algorithm for curves, then we use this machine to give another proof for the B´ezout’s theorem This content we follow from the article [1] by compute some explicit examples, we can see that, the Euclidean algorithm seems to be more helpful in calculate the intersection multiplicities 41 Bibliography [1] Jan Hilmar and Chris Smyth, 2010, Euclid Meets Bezout: Intersecting Algebraic Plane Curves with the Euclidean Algorithm , American Mathematical Monthly, vol 117, (3), 250-260 [2] Robert J Walker, 1950, Algebraic Curves, Princeton University Press, Princeton, (3), 50-62 42 ... 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. .. Hilmar and Chris Smyth finished their Bachelor thesis le thi trang 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