Density Of Images Of Polynomials With Zero Constant Terms On Upper Triangular Matrix Algebra And Waring Proble

The study of the image of polynomials on noncommutative algebras has been a rela-tively active research area in recent years.It is closely related to the theory of polynomial identity and the structure of algebra.Following the establishment of the theory of poly-nomial identity,people turned their interest to the study of the image of polynomials on non-commutative algebras,and have obtained a series of results.Similarly,the study of image of words on groups has also obtained some important results.The study of the image of polynomials originated from the following old and famous Lvov-Kaplansky conjecture.Lvov-Kaplansky conjecture.Let n≥2.Let F be a field.Let M_n（F）be the algebra of all n×n matrices over F.Let p（x₁,...,x_m）be a multilinear polynomial over F.Then the image of p on M_n（F）is a vector space.Up to now,the Lvov-Kaplansky conjecture has only held for the 2×2 matrix algebra over a quadratically closed field.However,it has not been solved for the 3×3 matrix algebra.In attempts to approach the Lvov-Kaplansky conjecture,some variations of it have been studied extensively.For example,many results on the image of multilinear polyno-mials on Lie algebra,Jordan algebra,Weyl algebra,graded algebra,quaternion algebra,upper triangular matrix algebra,strictly upper triangular matrix algebra and other algebras have been obtained.Upper triangular matrix algebras are important subalgebras of matrix algebras,and some arguments in the study of the image of polynomials on upper triangular matrix al-gebras may be applied to the study of the Lvov-Kaplansky conjecture.In 2019,Fagundes and Mello proposed the following conjecture:Fagundes-Mello conjecture.Let n≥2.Let F be a field.Let T_n（F）be the algebra of all n×n upper triangular matrices over F.Let p（x₁,...,x_m）be a multilinear polynomial over F.Then the image of p on T_n（F）is a vector space.At present,some important progress on the conjecture has been obtained.For exam-ple,in 2022,Luo and Wang proved that the conjecture holds when the cardinality of the field is sufficiently large.In 2023,Fagundes and Koshlukov studied the image of graded multilinear polynomials on graded upper triangular matrix algebras.In 2022,Gargate and Mello proposed the following problem:Gargate-Mello problem.Let F be an algebraically closed field.Let T_n（F）be the algebra of all n×n upper triangular matrices over F.Let p（x₁,...,x_m）be a completely homogeneous polynomial over F.Let r be the order of p.Is the image of p on T_n（F）a dense subset of T_n（F）^（r-1）（with respect to the Zariski topology）?The problem has been solved in the case of n=2,3,4.However the problem is still open in the case of n≥5 and 1<r<n-1.The Waring problem originated from a classical problem in number theory proposed by Waring in 1770.In the recent years,some results on the Waring problem for the image of polynomials on algebras have been obtained.For example,in 2023,Panja and Prasad initiated the study of the Waring problem for the image of polynomials with zero constant term on upper triangular matrix algebras.They proposed the following conjecture:Panja-Prasad conjecture.Let n≥4 be an integer.Let F be an algebraically closed field.Let T_n（F）be the upper triangular matrix algebra of all n×n matrices over F.Let p（x₁,...,x_m）be a polynomial with zero constant term in non-commutative variables over F.Suppose that 1<r<n-1,where r is the order of p.Then p（T_n（F））+p（T_n（F））=T_n（F）^（r-1）.The main goal of this paper is to solve both Gargate-Mello problem and Panja-Prasad conjecture.一、Solve the Gargate-Mello problem.We shall investigate the density of the image of polynomials with zero constant term on upper triangular matrix algebras.More precisely,we shall prove the following result:Main result 1.Let n≥2 and m≥1 be integers.Let F be an algebraically closed field.Let T_n（F）be the algebra of all n×n upper triangular matrices over F.Let p（x₁,...,x_m）be a polynomial with zero constant term in non-commutative variables over F.Let r be the order of p.Then the image of p on T_n（F）is a dense subset of T_n（F）^（r-1）（with respect to the Zariski topology）.Since a completely homogeneous polynomial is a polynomial with zero constant term,it is clear that Main result 1 completely solves the Gargate-Mello problem.二、Solve the Panja-Prasad conjecture.We shall investigate the Waring problem for the image of polynomials with zero constant term on upper triangular matrix algebras over an infinite field.More precisely,we shall prove the following result:Main result 2.Let n≥4 be an integer.Let F be an infinite field.Let T_n（F）be the algebra of all n×n upper triangular matrices over F.Let p（x₁,...,x_m）be a polynomial with zero constant term in non-commutative variables over F.Suppose that 1<r<n-1,where r is the order of p.Then p（T_n（F））+p（T_n（F））=T_n（F）^（r-1）.In particular,if r=n-2,then p（T_n（F））=T_n（F）^（n-3）.Since an algebraically closed field is an infinite field,it is clear that Main result 2completely solves the Panja-Prasad conjecture.In order to prove the above two results,we shall use some well-known arguments,such as the functional property of polynomials over an infinite field,the order of poly-nomials,and the density of Zariski topology.At the same time,we create some new arguments,such as the expression for the image of polynomials with zero constant term on the upper triangular matrix algebras,the compatibility of subsets in the field,adjoint polynomials,recursive polynomials.In view of Main result 2,we see that the Waring problem for the image of polyno-mials with zero constant term on upper triangular matrix algebras over an infinite field has been solved.Naturally,the discussion of the Waring problem for the image of poly-nomials with zero constant term on upper triangular matrix algebras over a finite field is a meaningful research topic.Morever,the Fagundes-Mello conjecture holds on an in-finite field,and naturally,it is also a meaningful research topic to discuss the image of multilinear polynomials on upper triangular matrix algebras over a finite field.