DIMACS REU 2012

General Information

Project Description

I will be looking into monic quadratic equations of the form X^2 + A*X + B = 0, where X, A, B and 0 are 2x2 matrixes with octonionic entries. It is known that when X, A, and B are complex matrices, there are 0, 1, 2, 3, 4, 5, 6 or infinitely many solutions to such an equation, and there are some techniques known that allow one to solve such equations with X, A, and B as quaternionic matrices. I will be attempting to come up with techniques for dealing with such equations when the matrices are octonionic, and to find similar restrictions on the number of roots of such equations.

Added later: Our project took several turns, and we also investigated equations such as nth-degree octonionic polynomials of a single variable with the coefficients on the left side of the indeterminate and with the coefficients in the span of 1 and a square root of -1 (e.g. isomorphic copies of complex numbers), and nth-degree quaternionic 2x2 matrix polynomials with “complex” coefficient matrices on the left of the indeterminate matrices.

Weekly Log

            Week 1:

I met with my mentor and we discussed the techniques for solving 2x2 matrix polynomial equations (particularly quadratics) over the complex numbers and over the quaternions. I read several articles about octonions and solved a few simpler examples of octonionic matrix equations, like X^2 = diag{1, 4}, and also proved a result about commutativity in the octonions.

Week 2:

My mentor and I met and continued reading through literature on octonions, particularly papers dealing with solutions of polynomial equations. There did not seem to be much about the roots of equations of the form x^2 + ax + b=0 for a, b, and x in O : One paper dealt with the roots of such equations when a and b were real numbers. So we started looking into automorphisms of octonions and came up with a method for “classifying” these types of octonionic equations. We also worked out several 2x2 quadratic quaternionic matrix equations with real coefficient matrices and found that if the equation had a finite number of complex 2x2 matrix solutions, then those were the only quaternionic matrix solutions. We began to attempt to prove whether or not this result holds in general.

Week 3:

We proved that a quadratic equation in one octonionic variable can have 1, 2 or infinitely many solutions. We became interested in knowing whether or not the factorization of a complex polynomial could tell you anything about the roots of such a polynomial if we considered the complex number i to be an octonionic square root of -1. **From now on, when we write “complex” in quotes, it refers to such an isomorphically embedded number in the octonions**

Week 4:

It turns out that a quadratic octonionic polynomial with “complex” coefficients only can have infinitely many solutions if the two “complex” roots are non-real “complex” conjugates. So we conjectured that if an nth-degree polynomial with “complex” coefficients has two non-real “complex” conjugate roots (which you can identify from the factorization of such a polynomial by thinking of the polynomial as actually being a standard complex polynomial (after application of an automorphism sending the square root of -1 to i)), then the polynomial has infinitely many octonionic roots. We started by proving this statement for “complex” polynomials with a quaternionic variable, under the assumption that the “complex” roots are all distinct.

Week 5:

We refined our proof to include the case of repeated roots. We also proved a lemma that allows us to extend the result from the quaternionic case to the octonionic: given an octonion u which cannot be written as “a + be1” (that is, a non-“complex” octonion), there exists an automorphism of the octonions which fixes the “complex” numbers of the form “a + be1” and takes u to a number of the form “a0 + a1e1 + a2e2” (that is, u is taken to an isomorphic embedding of the quaternions in the octonions).

Week 6:

We also proved that these “complex” polynomials ONLY can have infinitely many solutions if they have a pair of non-real “complex” conjugate roots. That is, there is only one cause of blow-up. We began to wonder how these results might translate into quaternionic and octonionic 2x2 matrix polynomials with “complex” matrix coefficients, and began looking at several examples which we knew to have infinitely many octonionic roots but only finitely many complex roots.

Week 7:

We went over a few examples of 2x2 quadratic matrix polynomials with “complex” coefficient matrices that we knew to have infinitely many roots, and found that if f(X) = X^2 + AX + B = 0 with A,B “complex” and det(f(tI)) has a pair of non-real “complex” conjugate roots, the equation f(X) =0 has infinitely many roots. We began to attempt to prove whether or not det(f(tI)) having such roots will always cause blow-up, for nth-degree polynomials, with n>1.

Week 8:

We noted that, via the application of an automorphism which fixes the non-imaginary part of a quaternion but moves around the other basis elements, that in a REAL coefficient matrix polynomial of degree n, if you can find a single non-real matrix solution, you can find infinitely many quaternionic matrix solutions. We conjectured that in an nth-degree 2x2 matrix polynomial f(X) with REAL matrix coefficients, if det(f(tI)) has a pair of non-real “complex” conjugate roots, then f(X) has a non-real “complex” solution, and hence infinitely many quaternionic solutions. We began working at proving this claim, and will continue to do so through the summer.

     
     

Presentations

 

Additional Information

·         My Mentors

o    Dr. Robert Wilson