For a finitely generated algebra A , let V be its finite-dimensional generating subspace. The

Alani Conner

Alani Conner

Answered question

2022-05-19

For a finitely generated algebra A, let V be its finite-dimensional generating subspace. Then A = n = 0 A n for A n = K + V + + V n . Let the function d V ( n ) = dim K ( A n ). Then the growth of A, i.e. G ( A ) will be defind as G ( A ) := G ( d V ).
I cannot understand why the growth of the commutative polynomial algebra K [ x 1 , , x d ] is polynomial of degree d; i.e. is P d and has a polynomial growth?

Answer & Explanation

Briana Grimes

Briana Grimes

Beginner2022-05-20Added 5 answers

An is the dimension of K + V + + V n which is the number of monomials x 1 t 1 x d t d with t 1 + + t n d. This is the number of d-tuples of nonnegative integers ( t 1 , , t d ) with sum n. That is the number of ( d + 1 )-tuples of nonnegative integers ( t 1 , , t d , t d + 1 ) with sum n (setting t d + 1 = n t 1 t d ). This number is ( n + d d ) by the "stars and bars" argument. Then for fixed d

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Commutative Algebra

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?