A Sampling of Symmetry and Solvability. Sarah Witherspoon Professor of Mathematics Texas A&M University

Transcription

1 A Sampling of Symmetry and Solvability Sarah Witherspoon Professor of Mathematics Texas A&M University

2 Polynomial Equations Examples: x 9 2x 4 5 = 0, x 2 y 3 3y 2 = 0, x 2 + y 2 + z 2 = 1 Solving polynomial equations has many applications, e.g. robotics and computer vision computer animation design of airplane wings font generation algorithms for matrix multiplication and many many more

3 Polynomial Equations Examples: x 9 2x 4 5 = 0, x 2 y 3 3y 2 = 0, x 2 + y 2 + z 2 = 1 Solving polynomial equations has many applications, e.g. robotics and computer vision computer animation design of airplane wings font generation algorithms for matrix multiplication and many many more

4 Polynomial Equations Examples: x 9 2x 4 5 = 0, x 2 y 3 3y 2 = 0, x 2 + y 2 + z 2 = 1 Solving polynomial equations has many applications, e.g. robotics and computer vision computer animation design of airplane wings font generation algorithms for matrix multiplication and many many more

5 Polynomial Equations Examples: x 9 2x 4 5 = 0, x 2 y 3 3y 2 = 0, x 2 + y 2 + z 2 = 1 Solving polynomial equations has many applications, e.g. robotics and computer vision computer animation design of airplane wings font generation algorithms for matrix multiplication and many many more

6 Polynomial Equations Examples: x 9 2x 4 5 = 0, x 2 y 3 3y 2 = 0, x 2 + y 2 + z 2 = 1 Solving polynomial equations has many applications, e.g. robotics and computer vision computer animation design of airplane wings font generation algorithms for matrix multiplication and many many more

7 Polynomial Equations Examples: x 9 2x 4 5 = 0, x 2 y 3 3y 2 = 0, x 2 + y 2 + z 2 = 1 Solving polynomial equations has many applications, e.g. robotics and computer vision computer animation design of airplane wings font generation algorithms for matrix multiplication and many many more

8 Polynomial Equations Examples: x 9 2x 4 5 = 0, x 2 y 3 3y 2 = 0, x 2 + y 2 + z 2 = 1 Solving polynomial equations has many applications, e.g. robotics and computer vision computer animation design of airplane wings font generation algorithms for matrix multiplication and many many more

9 Polynomial Equations Examples: x 9 2x 4 5 = 0, x 2 y 3 3y 2 = 0, x 2 + y 2 + z 2 = 1 Solving polynomial equations has many applications, e.g. robotics and computer vision computer animation design of airplane wings font generation algorithms for matrix multiplication and many many more

10 x+1944y+27444xy+9528xy x 2 y 93708xy x 2 y x 3 y 78816xy x 2 y x 3 y x 4 y y 5 x y 4 x y 3 x x 4 y x 5 y 6690y x y x y x y x y x 6 = 0 Hessian of a quintic with 8 ovals c Frank Sottile

11 courtesy of Herwig Hauser

12 History Babylon 2000 BC Solutions of the quadratic equation ax 2 + bx + c = 0 are given by the quadratic formula x = b ± b 2 4ac 2a What about the cubic equation ax 3 + bx 2 + cx + d = 0?

13 History Babylon 2000 BC Solutions of the quadratic equation ax 2 + bx + c = 0 are given by the quadratic formula x = b ± b 2 4ac 2a What about the cubic equation ax 3 + bx 2 + cx + d = 0?

14 History Babylon 2000 BC Solutions of the quadratic equation ax 2 + bx + c = 0 are given by the quadratic formula x = b ± b 2 4ac 2a What about the cubic equation ax 3 + bx 2 + cx + d = 0?

15 Gerolamo Cardano ( )

16 Cardano solved the cubic equation by reducing it to x 3 + cx + d = 0 Del Ferro had solved this case and gave the solution to Fior, who then challenged Tartaglia to a contest, inspiring Tartaglia to solve this himself, and he gave the solution to Cardano: x = 3 d 27 3 d 27 Ferrari solved the quartic equation similarly What about the quintic? No! Not solvable by radicals!

17 Cardano solved the cubic equation by reducing it to x 3 + cx + d = 0 Del Ferro had solved this case and gave the solution to Fior, who then challenged Tartaglia to a contest, inspiring Tartaglia to solve this himself, and he gave the solution to Cardano: x = 3 d 27 3 d 27 Ferrari solved the quartic equation similarly What about the quintic? No! Not solvable by radicals!

18 Cardano solved the cubic equation by reducing it to x 3 + cx + d = 0 Del Ferro had solved this case and gave the solution to Fior, who then challenged Tartaglia to a contest, inspiring Tartaglia to solve this himself, and he gave the solution to Cardano: x = 3 d 27 3 d 27 Ferrari solved the quartic equation similarly What about the quintic? No! Not solvable by radicals!

19 Cardano solved the cubic equation by reducing it to x 3 + cx + d = 0 Del Ferro had solved this case and gave the solution to Fior, who then challenged Tartaglia to a contest, inspiring Tartaglia to solve this himself, and he gave the solution to Cardano: x = 3 d 27 3 d 27 Ferrari solved the quartic equation similarly What about the quintic? No! Not solvable by radicals!

20 Cardano solved the cubic equation by reducing it to x 3 + cx + d = 0 Del Ferro had solved this case and gave the solution to Fior, who then challenged Tartaglia to a contest, inspiring Tartaglia to solve this himself, and he gave the solution to Cardano: x = 3 d 27 3 d 27 Ferrari solved the quartic equation similarly What about the quintic? No! Not solvable by radicals!

21 Cardano solved the cubic equation by reducing it to x 3 + cx + d = 0 Del Ferro had solved this case and gave the solution to Fior, who then challenged Tartaglia to a contest, inspiring Tartaglia to solve this himself, and he gave the solution to Cardano: x = 3 d 27 3 d 27 Ferrari solved the quartic equation similarly What about the quintic? No! Not solvable by radicals!

22 Cardano solved the cubic equation by reducing it to x 3 + cx + d = 0 Del Ferro had solved this case and gave the solution to Fior, who then challenged Tartaglia to a contest, inspiring Tartaglia to solve this himself, and he gave the solution to Cardano: x = 3 d 27 3 d 27 Ferrari solved the quartic equation similarly What about the quintic? No! Not solvable by radicals!

23 Cardano solved the cubic equation by reducing it to x 3 + cx + d = 0 Del Ferro had solved this case and gave the solution to Fior, who then challenged Tartaglia to a contest, inspiring Tartaglia to solve this himself, and he gave the solution to Cardano: x = 3 d 27 3 d 27 Ferrari solved the quartic equation similarly What about the quintic? No! Not solvable by radicals!

24 Cardano solved the cubic equation by reducing it to x 3 + cx + d = 0 Del Ferro had solved this case and gave the solution to Fior, who then challenged Tartaglia to a contest, inspiring Tartaglia to solve this himself, and he gave the solution to Cardano: x = 3 d 27 3 d 27 Ferrari solved the quartic equation similarly What about the quintic? No! Not solvable by radicals!

25 Cardano solved the cubic equation by reducing it to x 3 + cx + d = 0 Del Ferro had solved this case and gave the solution to Fior, who then challenged Tartaglia to a contest, inspiring Tartaglia to solve this himself, and he gave the solution to Cardano: x = 3 d 27 3 d 27 Ferrari solved the quartic equation similarly What about the quintic? No! Not solvable by radicals!

26 Cardano solved the cubic equation by reducing it to x 3 + cx + d = 0 Del Ferro had solved this case and gave the solution to Fior, who then challenged Tartaglia to a contest, inspiring Tartaglia to solve this himself, and he gave the solution to Cardano: x = 3 d 27 3 d 27 Ferrari solved the quartic equation similarly What about the quintic? No! Not solvable by radicals!

27 Évariste Galois ( )

28 Galois used symmetry in polynomial equations, e.g. z 6 = i i i i

29 Texas A&M math circle: symmetry and Rubik s cube

30 Using symmetry, Galois proved that some quintics ax 5 + bx 4 + cx 3 + dx 2 + ex + f = 0 are not solvable by radicals i.e. they do not have solutions using only radicals (square roots, cube roots,... ) and multiplication/division/addition/subtraction of a, b, c, d, e, f Same for polynomial equations of degree higher than 5 Question: How do we solve them, then? Some answers: use the more general hypergeometric functions numerically, with computers

31 Using symmetry, Galois proved that some quintics ax 5 + bx 4 + cx 3 + dx 2 + ex + f = 0 are not solvable by radicals i.e. they do not have solutions using only radicals (square roots, cube roots,... ) and multiplication/division/addition/subtraction of a, b, c, d, e, f Same for polynomial equations of degree higher than 5 Question: How do we solve them, then? Some answers: use the more general hypergeometric functions numerically, with computers

32 Using symmetry, Galois proved that some quintics ax 5 + bx 4 + cx 3 + dx 2 + ex + f = 0 are not solvable by radicals i.e. they do not have solutions using only radicals (square roots, cube roots,... ) and multiplication/division/addition/subtraction of a, b, c, d, e, f Same for polynomial equations of degree higher than 5 Question: How do we solve them, then? Some answers: use the more general hypergeometric functions numerically, with computers

33 Using symmetry, Galois proved that some quintics ax 5 + bx 4 + cx 3 + dx 2 + ex + f = 0 are not solvable by radicals i.e. they do not have solutions using only radicals (square roots, cube roots,... ) and multiplication/division/addition/subtraction of a, b, c, d, e, f Same for polynomial equations of degree higher than 5 Question: How do we solve them, then? Some answers: use the more general hypergeometric functions numerically, with computers

34 Using symmetry, Galois proved that some quintics ax 5 + bx 4 + cx 3 + dx 2 + ex + f = 0 are not solvable by radicals i.e. they do not have solutions using only radicals (square roots, cube roots,... ) and multiplication/division/addition/subtraction of a, b, c, d, e, f Same for polynomial equations of degree higher than 5 Question: How do we solve them, then? Some answers: use the more general hypergeometric functions numerically, with computers

35 Using symmetry, Galois proved that some quintics ax 5 + bx 4 + cx 3 + dx 2 + ex + f = 0 are not solvable by radicals i.e. they do not have solutions using only radicals (square roots, cube roots,... ) and multiplication/division/addition/subtraction of a, b, c, d, e, f Same for polynomial equations of degree higher than 5 Question: How do we solve them, then? Some answers: use the more general hypergeometric functions numerically, with computers

36 Computing at Texas A&M

37 Mathematics Students at Texas A&M Math Club (Pi Mu Epsilon) AMUSE (Applied Mathematics Undergraduate Seminar) Aggie Actuaries Honors Program Research Experiences

38 Texas A&M students and faculty at Mathfest

39 Texas A&M Mathematics Graduates Careers in teaching actuarial science consulting firms or technical companies government labs/agencies Many continue studies at graduate/professional schools