The Images Of Multilinear Polynomials On Upper Triangular Matrix Algebras

The study of the image of polynomials on noncommutative algebras is an active research direction in noncommutative algebras.The study of image of polynomials on noncommutative algebras is closely related to the theory of polynomial identity（PI-theory）on noncommutative algebras.After the theory of polynomial identity was established,people began to pay attentions to the image of polynomials on noncommutative algebras.At present,a large number of the results on the image of polynomials on noncommutative algebras have been obtained.Parallel to the image of a polynomial on a noncommutative algebras,the study of the image of words on groups is also a research topic of group theory.Moreover,the image of words on groups have been applied to some important problems on groups.The study of the image of polynomials on noncommutative algebras originated from the following old and famous Lvov-Kaplansky conjecture:Lvov-Kaplansky conjecture The image of a multilinear polynomial on matrix algebras over a field is a vector space.Up to now,the conjecture is affirmative only in either multilinear polynomials with low degree or 2 × 2 matrix algebras.In other words,the conjecture is far from being solved.While studying the Lvov-Kaplansky conjecture,people have also studied its variations.At present,many results on the images of polynomials on strictly upper triangular matrix algebra,upper triangular matrix algebra,Lie（Jordan）algebra,quaternion algebra,graded matrix algebra and other algebras have been obtained.Upper triangular matrix algebras are important subalgebras of matrix algebras.Studying the image of polynomials on upper triangular matrix algebras is not only of its own significance,but also of certain reference significance for studying Lvov-Kaplansky conjecture.In 2019,Fagundes and Mello proposed the following conjecture:Fagundes-Mello conjecture The image of a multilinear polynomial on upper triangular matrix algebras over a field is a vector space.There exists some results on the conjecture.For example,in 2022,Gargate and Mello proved that the Fagundes-Mello conjecture is true on an infinite field.In the present paper,under a mild condition on the ground field we shall prove that the Fagundes-Mello conjecture is true.More precisely,the main result of the present paper is:Main result Let n≥2 be an integer.Let m>1 be an integer.Let K be a field.Let UT_n be a n × n upper triangular matrix algebra over K.Let f（x1,...,xm）be a nonzero multilinear polynomial in non-commutative variables over K.Suppose that K contains at least n（n-1）/2 elements.We have that f（UT_n）either {0}.UT_n or UT_n^k.for some integer 0≤k≤m/2-1.In particular,f（UT_n）is a vector subspace of UT_n.As far as we know,this is the best result on the Fagundes-Mello conjecture.We shall define the β-index of multilinear polynomials.By the β-index and the triangular algebra we shall prove the main result.As applications of the main results we shall give the three results.Firstly,we use theβ-index and the main results to give a sufficient condition for the image of a multilinear polynomial on matrix algebra to contain two distinct non-zero vector spaces.Secondly,we shall give a criterion that a multilinear polynomial is not an identity on matrix algebras.Finally,we shall give a description of the image of Jordan multilinear polynomials on Jordan upper triangular matrix algebras.The β-index of multilinear polynomials is a original definition of the present paper,which plays a key role in proving the main result.We believe that this definition also has some reference significance for the study of the more complex Lvov-Kaplansky conjecture.