In this paper we study identities and images of polynomials on null-filiform
Leibniz algebras. If Lnβ is an n-dimensional null-filiform Leibniz algebra,
we exhibit a finite minimal basis for \mbox{Id}(L_n), the polynomial
identities of Lnβ, and we explicitly compute the images of multihomogeneous
polynomials on Lnβ. We present necessary and sufficient conditions for the
image of a multihomogeneous polynomial f to be a subspace of Lnβ. For the
particular case of multilinear polynomials, we prove that the image is always a
vector space, showing that the analogue of the L'vov-Kaplansky conjecture holds
for Lnβ. We also prove similar results for an analog of null-filiform Leibniz
algebras in the infinite-dimensional case.Comment: 13 pages; comments are welcom