# Revision history [back]

This is the difference between dense and sparse representations. In the first case, the polynomial is x^8 + x^7 - x^5 - x^4 - x^3 + x + 1 so the list of coefficients is [1, 1, 0, -1, -1, -1, 0, 1, 1]. In the second case, the polynomial is x^40000 - x^30000 + x^20000 - x^10000 + 1 so instead of storing the first 40001 coefficients, it is wiser to only store the nonzero ones. For this, Sage use a dictionary that tells that x^0 has coefficient 1, x^1000 has coefficient -1 and so on.

This is the difference between dense and sparse representations. In the first case, the polynomial is x^8 + x^7 - x^5 - x^4 - x^3 + x + 1 so the list of coefficients is [1, 1, 0, -1, -1, -1, 0, 1, 1]. In the second case, the polynomial is x^40000 - x^30000 + x^20000 - x^10000 + 1 so instead of storing the first 40001 coefficients, it is wiser to only store the nonzero ones. For this, Sage use a dictionary that tells that x^0 has coefficient 1, x^1000 has coefficient -1 , and so on.

As written in the doc, you can get the polynomials as follows:

sage: R = QQ['x']
sage: R
Univariate Polynomial Ring in x over Rational Field
sage: R(cyclotomic_coeffs(30))
x^8 + x^7 - x^5 - x^4 - x^3 + x + 1
sage: R(cyclotomic_coeffs(10^5))
x^40000 - x^30000 + x^20000 - x^10000 + 1


This is the difference between dense and sparse representations. In the first case, the polynomial is x^8 + x^7 - x^5 - x^4 - x^3 + x + 1 so it can be represented as the list of coefficients is its first 9 coefficients: [1, 1, 0, -1, -1, -1, 0, 1, 1]. In the second case, the polynomial is x^40000 - x^30000 + x^20000 - x^10000 + 1 so instead of storing the first 40001 coefficients, it is wiser to only store the nonzero ones. For this, Sage use a dictionary that tells that x^0 has coefficient 1, x^1000 has coefficient -1, and so on.

As written in the doc, you can get the polynomials as follows:

sage: R = QQ['x']
sage: R
Univariate Polynomial Ring in x over Rational Field
sage: R(cyclotomic_coeffs(30))
x^8 + x^7 - x^5 - x^4 - x^3 + x + 1
sage: R(cyclotomic_coeffs(10^5))
x^40000 - x^30000 + x^20000 - x^10000 + 1