# 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.

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What is sparse representation?

( 2024-01-27 04:08:39 +0200 )edit

from `PolynomialRing? :

• "sparse" -- bool: whether or not elements are sparse. The default is a dense representation ("sparse=False") for univariate rings and a sparse representation ("sparse=True") for multivariate rings.
( 2024-01-27 07:57:04 +0200 )edit

Could you give us a concrete example ? A code snippet would do...

( 2024-01-27 12:59:08 +0200 )edit

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.

( 2024-01-27 14:55:05 +0200 )edit

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.

( 2024-01-29 12:01:29 +0200 )edit