# finding a basis of a set of polynomials in sage

The task I want to do is to compute a basis of the k-th order partial derivative space of a polynomial f. By k-th order partial derivative space of a polynomial f I mean the vector space formed by all the k-th order derivatives of f. So one way that I know is to compute all the partial derivatives and then treat each polynomial as a vectors and then compute basis. But is there a way to do it in the sparse representation.

What is sparse representation?

@Max Alekseyev :

from `

`PolynomialRing?`

:Could you give us a

concreteexample ? A code snippet would do...I do not see how sparseness of polynomial representation is relevant here. Derivatives can be taken in spare form and vectors can be represented in sparse form as well.

To compute the basis of the k-th order partial derivative space of a polynomial in a sparse representation, one can employ symbolic mathematics libraries like SymPy. Begin by symbolically representing the original polynomial and expressing it in sparse form with non-zero coefficients. Utilize sparse polynomial manipulation techniques to derive the k-th order partial derivative in a sparse representation. Filter out non-zero terms and treat each resulting polynomial as a vector. Finally, compute the basis of the space using sparse linear https://www.igmguru.com/salesforce/salesforce-cpq-training/ (algebra) techniques for optimization. This approach provides an efficient way to handle high-degree polynomials while taking advantage of sparse representations.