Narrow Results

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 × 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). This conjecture was proved for n = 2 when K is closed under quadratic extensions. In this paper, the conjecture is verified for K = ℝ and n = 2, also for semi-homogeneous polynomials p, with a partial solution for an arbitrary field K.

Let $p$ be a multilinear polynomial in several noncommuting 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 a vector space. In this paper, we settle the analogous conjecture for a quaternion algebra.

Kaplansky conjectured that if two positive-definite ternary quadratic forms have perfectly identical representations over $\mathbb{Z}$, they are equivalent over $\mathbb{Z}$ or constant multiples of regular forms, or is included in either of two families parameterized by ${ℝ}^2$. Our results aim to clarify the limitations imposed to such a pair by computational and theoretical approaches. First, the result of an exhaustive search for such pairs of integral quadratic forms is presented in order to provide a concrete version of the Kaplansky conjecture. The obtained list contains a small number of non-regular forms that were confirmed to have the identical representations up to 3,000,000 by computation. However, a strong limitation on the existence of such pairs is still observed, regardless of whether the coefficient field is $\mathbb{Q}$ or $ℝ$. Second, we prove that if two pairs of ternary quadratic forms have the identical simultaneous representations over $\mathbb{Q}$, their constant multiples are equivalent over $\mathbb{Q}$. This was motivated by the question why the other families were not detected in the search. In the proof, the parametrization of quartic rings and their resolvent rings by Bhargava is used to discuss pairs of ternary quadratic forms.