# The images of non-commutative polynomials evaluated on 2 x 2 matrices over the real numbers

Seminar

Speaker

Sergey Malev (Bar-Ilan University)

Date

13/11/2013 - 10:30Add to Calendar

`2013-11-13 10:30:00``2013-11-13 10:30:00``The images of non-commutative polynomials evaluated on 2 x 2 matrices over the real numbers``Let p be a multilinear polynomial in several non-commuting variables with coefficients in an arbitrary field K. Kaplansky conjectured that for any n, the image of p evaluated on the set M_n(K) of n-by-n matrices is either zero, or the set of scalar matrices, or the set sl_n(K) of matrices of trace 0, or all of M_n(K). I prove the conjecture when K is the field of real numbers and n=2, and give a partial solution for an arbitrary field K.``אוניברסיטת בר-אילן - Department of Mathematics``mathoffice@math.biu.ac.il``Asia/Jerusalem``public`Abstract

Let p be a multilinear polynomial in several non-commuting

variables with coefficients in an arbitrary field K. Kaplansky

conjectured that for any n, the image of p evaluated on the

set M_n(K) of n-by-n matrices is either zero, or the set of

scalar matrices, or the set sl_n(K) of matrices of trace 0, or

all of M_n(K). I prove the conjecture when K is the field of real numbers and

n=2, and give a partial solution for

an arbitrary field K.

Last Updated Date : 06/11/2013