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The system of tensor equations (TEs) has received much considerable attentionin the recent literature. In this paper, we consider a class of generalizedtensor equations (GTEs). An important difference between GTEs and TEs is thatGTEs can be regarded as a system of non-homogenous polynomial equations,whereas TEs is a homogenous one. Such a difference usually makes thetheoretical and algorithmic results tailored for TEs not necessarily applicableto GTEs. To study properties of the solution set of GTEs, we first introduce anew class of so-named ${\rm Z}^+$-tensor, which includes the set of allP-tensors as its proper subset. With the help of degree theory, we prove thatthe system of GTEs with a leading coefficient ${\rm Z}^+$-tensor has at leastone solution for any right-hand side vector. Moreover, we study the local errorbounds under some appropriate conditions. Finally, we employ aLevenberg-Marquardt algorithm to find a solution to GTEs and report somepreliminary numerical results.

Generalized tensor equations with leading structured tensors