QUALIFYING EXAMINATION Harvard University Department of Mathematics Tuesday JanuaryDayLet K = C(x) be the field of rational functions in one variable over C, and

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MHF 4U Unit 2 –Rational Functions– Outline

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MHF 4U Unit 2 –Rational Functions– Outline Specific Expectations Day

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Automatic congruences for diagonals of rational functions Eric Rowland Universit´e du Qu´ebec a` Montr´eal, Montr´eal, Canada Reem Yassawi∗ Trent University, Peterborough, Canada

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Sum-product estimates for rational functions Boris Bukh MayA story of a romance between graphs and arithmetic

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Unit 2: Rational Functions MHF4U Lesson Outline

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Geometric Rank Functions and Rational Points on Curves Eric Katz (University of Waterloo) joint with David Zureick-Brown (Emory University) January 9, 2013

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Minimal Factorizations of Rational Matrix Functions in Terms of Polynomial Models T. Shamir and P. A. Fuhrmann Department of Mathematics and Computer Science Ben Gurion University of the Negev Beer Sheva, Israel

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Tangent Spaces of Rational Matrix Functions* U. Helmke Universitiit Wiirzburg Institute ftir Mathematik Wiirzbur-g, Germany and

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The Enhanced Pathway allows for a motivated student to complete the necessary course work to take Calculus in grade twelve starting with Enhanced Algebra I in grade nine (followed by Enhanced Geometry in grade ten and En

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• Trigonometry
• Elementary mathematics
• Functions and mappings
• Polynomial
• Quadratic equation
• Trigonometric functions
• Complex number
• Generalized trigonometry
• Rational function
• Mathematics
• Algebra
• Equations