Daniela Mueller (UCD)

will speak on

Some Results on Linearized Trinomials that Split Completely

Time: 2:00PM
Date: Thu 21st November 2019
Location: Seminar Room SCN 1.25 [map]

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Abstract: Linearized polynomials over finite fields have been much studied over the last several decades. Recently there has been a renewed interest in linearized polynomials because of new connections to coding theory and finite geometry. We consider the problem of calculating the rank or nullity of a linearized polynomial $L(x)$ from its coefficients. The rank and nullity of $L(x)$ are the rank and nullity of the associated $F_q$-linear map $F_{q^n} -> F_{q^n}$. McGuire and Sheekey defined a d x d matrix $A_L$ with the property that nullity($L$) = nullity($A_L-I$). We present some consequences of this result for the ranks of particular types of linearized trinomials. For example, we are able to generalize a result of Csajbok, Marino, Polverino and Zhou which states that $ax + bx^q + cx^{q^3}$ (where $a,b,c$ are in $F_{q^7}$) cannot have $q^3$ roots in $F_{q^7}$ if $q$ is odd.

(This talk is part of the Algebra and Number Theory series.)

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