Bézout's Theorem
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.
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