Bézout’s Theorem Jennifer Li Department of Mathematics University of Massachusetts Amherst, MA ¡ No common components How many intersection points are there? Motivation f (x,y) g(x,y) Jennifer Li (University of Massachusetts) Bézout’s Theorem November 30, 2016 2 / 80 How many intersection points are there? Motivation f (x,y) ¡ No common components g(x,y) Jennifer Li (University of Massachusetts) Bézout’s Theorem November 30, 2016 2 / 80 Motivation f (x,y) ¡ No common components g(x,y) How many intersection points are there? Jennifer Li (University of Massachusetts) Bézout’s Theorem November 30, 2016 2 / 80 Algebra Geometry { set of solutions } { intersection points of curves } Motivation f (x,y) = 0 g(x,y) = 0 Jennifer Li (University of Massachusetts) Bézout’s Theorem November 30, 2016 3 / 80 Geometry { intersection points of curves } Motivation f (x,y) = 0 g(x,y) = 0 Algebra { set of solutions } Jennifer Li (University of Massachusetts) Bézout’s Theorem November 30, 2016 3 / 80 Motivation f (x,y) = 0 g(x,y) = 0 Algebra Geometry { set of solutions } { intersection points of curves } Jennifer Li (University of Massachusetts) Bézout’s Theorem November 30, 2016 3 / 80 Motivation f (x,y) = 0 g(x,y) = 0 How many intersections are there? Jennifer Li (University of Massachusetts) Bézout’s Theorem November 30, 2016 4 / 80 Motivation f (x,y) = 0 g(x,y) = 0 How many intersections are there? Jennifer Li (University of Massachusetts) Bézout’s Theorem November 30, 2016 5 / 80 Motivation f (x,y) = 0 g(x,y) = 0 How many intersections are there? Jennifer Li (University of Massachusetts) Bézout’s Theorem November 30, 2016 6 / 80 f (x): polynomial in one variable 2 n f (x) = a0 + a1x + a2x +⋯+ anx Example. f (x) = 1 + 8x + x2 + 5x3 + 12x5 The degree of a polynomial f (x) is the largest power of x with nonzero coefficient. Example. f (x) = 1 + 8x + x2 + 5x3 + 12x5 The degree of f (x) is 5. Polynomials: One Variable Jennifer Li (University of Massachusetts) Bézout’s Theorem November 30, 2016 7 / 80 Example. f (x) = 1 + 8x + x2 + 5x3 + 12x5 The degree of a polynomial f (x) is the largest power of x with nonzero coefficient. Example. f (x) = 1 + 8x + x2 + 5x3 + 12x5 The degree of f (x) is 5. Polynomials: One Variable f (x): polynomial in one variable 2 n f (x) = a0 + a1x + a2x +⋯+ anx Jennifer Li (University of Massachusetts) Bézout’s Theorem November 30, 2016 7 / 80 The degree of a polynomial f (x) is the largest power of x with nonzero coefficient. Example. f (x) = 1 + 8x + x2 + 5x3 + 12x5 The degree of f (x) is 5. Polynomials: One Variable f (x): polynomial in one variable 2 n f (x) = a0 + a1x + a2x +⋯+ anx Example. f (x) = 1 + 8x + x2 + 5x3 + 12x5 Jennifer Li (University of Massachusetts) Bézout’s Theorem November 30, 2016 7 / 80 Example. f (x) = 1 + 8x + x2 + 5x3 + 12x5 The degree of f (x) is 5. Polynomials: One Variable f (x): polynomial in one variable 2 n f (x) = a0 + a1x + a2x +⋯+ anx Example. f (x) = 1 + 8x + x2 + 5x3 + 12x5 The degree of a polynomial f (x) is the largest power of x with nonzero coefficient. Jennifer Li (University of Massachusetts) Bézout’s Theorem November 30, 2016 7 / 80 Polynomials: One Variable f (x): polynomial in one variable 2 n f (x) = a0 + a1x + a2x +⋯+ anx Example. f (x) = 1 + 8x + x2 + 5x3 + 12x5 The degree of a polynomial f (x) is the largest power of x with nonzero coefficient. Example. f (x) = 1 + 8x + x2 + 5x3 + 12x5 The degree of f (x) is 5. Jennifer Li (University of Massachusetts) Bézout’s Theorem November 30, 2016 7 / 80 f (x,y): polynomial in two variables 2 2 n n f (x,y) = a00 + a10x + a01y + a20x + a11xy + a02y +⋯+ an0x +⋯+ a0ny i j a i j x y Polynomials: Two Variables Jennifer Li (University of Massachusetts) Bézout’s Theorem November 30, 2016 8 / 80 Polynomials: Two Variables f (x,y): polynomial in two variables 2 2 n n f (x,y) = a00 + a10x + a01y + a20x + a11xy + a02y +⋯+ an0x +⋯+ a0ny i j a i j x y Jennifer Li (University of Massachusetts) Bézout’s Theorem November 30, 2016 8 / 80 Polynomials: Two Variables f (x,y): polynomial in two variables 2 2 n n f (x,y) = a00 + a10x + a01y + a20x + a11xy + a02y +⋯+ an0x +⋯+ a0ny i j a ij x y Jennifer Li (University of Massachusetts) Bézout’s Theorem November 30, 2016 9 / 80 Polynomials: Two Variables f (x,y): polynomial in two variables 2 2 n n f (x,y) = a00 + a10x + a01y + a20x + a11xy + a02y +⋯+ an0x +⋯+ a0ny i j a ij x y Jennifer Li (University of Massachusetts) Bézout’s Theorem November 30, 2016 10 / 80 Polynomials: Two Variables f (x,y): polynomial in two variables 2 2 n n f (x,y) = a00 + a10x + a01y + a20x + a11xy + a02y +⋯+ an0x +⋯+ a0ny i + j = 0 i + j = 1 i + j = 2 ⋮ i + j = n Jennifer Li (University of Massachusetts) Bézout’s Theorem November 30, 2016 11 / 80 Example. f (x,y) = 5x2 + 8xy + 12y2 − 4x + y + 13 is a polynomial of degree 2. Degree of a Polynomial The degree of a polynomial f (x,y) is the largest sum of powers of x and y. Jennifer Li (University of Massachusetts) Bézout’s Theorem November 30, 2016 12 / 80 Degree of a Polynomial The degree of a polynomial f (x,y) is the largest sum of powers of x and y. Example. f (x,y) = 5x2 + 8xy + 12y2 − 4x + y + 13 is a polynomial of degree 2. Jennifer Li (University of Massachusetts) Bézout’s Theorem November 30, 2016 12 / 80 Polynomials over a field k 2 n f (x) = a0 + a1x + a2x +⋯+ anx ⊛ In Calculus I: usually over field k = R ⊛ Means ai all belong to R Polynomials over R Jennifer Li (University of Massachusetts) Bézout’s Theorem November 30, 2016 13 / 80 ⊛ In Calculus I: usually over field k = R ⊛ Means ai all belong to R Polynomials over R Polynomials over a field k 2 n f (x) = a0 + a1x + a2x +⋯+ anx Jennifer Li (University of Massachusetts) Bézout’s Theorem November 30, 2016 13 / 80 Polynomials over R Polynomials over a field k 2 n f (x) = a0 + a1x + a2x +⋯+ anx ⊛ In Calculus I: usually over field k = R ⊛ Means ai all belong to R Jennifer Li (University of Massachusetts) Bézout’s Theorem November 30, 2016 13 / 80 Polynomials over R Polynomials over a field k 2 n f (x) = a0 + a1x + a2x +⋯+ anx ⊛ In Calculus I: usually over field k = R ⊛ Means ai’s all belong to R Jennifer Li (University of Massachusetts) Bézout’s Theorem November 30, 2016 14 / 80 2 n f (x) = a0 + a1x + a2x +⋯+ anx ⊛ Means ai’s all belong to C Polynomials over C Polynomials over field C Jennifer Li (University of Massachusetts) Bézout’s Theorem November 30, 2016 15 / 80 Polynomials over C Polynomials over field C 2 n f (x) = a0 + a1x + a2x +⋯+ anx ⊛ Means ai’s all belong to C Jennifer Li (University of Massachusetts) Bézout’s Theorem November 30, 2016 15 / 80 Numbers of form a + bi where ⊛ numbers a and b are in R ⊛ the symbol i is the imaginary number with property i 2 = −1 a ∶ the real part b ∶ the imaginary part ⊛ every real number is of the form a + 0 i Complex Numbers Jennifer Li (University of Massachusetts) Bézout’s Theorem November 30, 2016 16 / 80 where ⊛ numbers a and b are in R ⊛ the symbol i is the imaginary number with property i 2 = −1 a ∶ the real part b ∶ the imaginary part ⊛ every real number is of the form a + 0 i Complex Numbers Numbers of form a + bi Jennifer Li (University of Massachusetts) Bézout’s Theorem November 30, 2016 16 / 80 ⊛ the symbol i is the imaginary number with property i 2 = −1 a ∶ the real part b ∶ the imaginary part ⊛ every real number is of the form a + 0 i Complex Numbers Numbers of form a + bi where ⊛ numbers a and b are in R Jennifer Li (University of Massachusetts) Bézout’s Theorem November 30, 2016 16 / 80 a ∶ the real part b ∶ the imaginary part ⊛ every real number is of the form a + 0 i Complex Numbers Numbers of form a + bi where ⊛ numbers a and b are in R ⊛ the symbol i is the imaginary number with property i 2 = −1 Jennifer Li (University of Massachusetts) Bézout’s Theorem November 30, 2016 16 / 80 b ∶ the imaginary part ⊛ every real number is of the form a + 0 i Complex Numbers Numbers of form a + bi where ⊛ numbers a and b are in R ⊛ the symbol i is the imaginary number with property i 2 = −1 a ∶ the real part Jennifer Li (University of Massachusetts) Bézout’s Theorem November 30, 2016 16 / 80 ⊛ every real number is of the form a + 0 i Complex Numbers Numbers of form a + bi where ⊛ numbers a and b are in R ⊛ the symbol i is the imaginary number with property i 2 = −1 a ∶ the real part b ∶ the imaginary part Jennifer Li (University of Massachusetts) Bézout’s Theorem November 30, 2016 16 / 80 Complex Numbers Numbers of form a + bi where ⊛ numbers a and b are in R ⊛ the symbol i is the imaginary number with property i 2 = −1 a ∶ the real part b ∶ the imaginary part ⊛ every real number is of the form a + 0 i Jennifer Li (University of Massachusetts) Bézout’s Theorem November 30, 2016 16 / 80 C R Algebraic Closure Algebraically closed? Jennifer Li (University of Massachusetts) Bézout’s Theorem November 30, 2016 17 / 80 R Algebraic Closure Algebraically closed? C Jennifer Li (University of Massachusetts) Bézout’s Theorem November 30, 2016 17 / 80 Algebraic Closure Algebraically closed? C R Jennifer Li (University of Massachusetts) Bézout’s Theorem November 30, 2016 17 / 80 For any nonconstant polynomial f (x) over C, we can find a number r in C such that f (r) = 0.