Defining family of multivariable polynomials

DCL
13 ●1 ●5

Brand new to Sage here and trying to define a family of polynomials indexed by natural numbers. In particular, I'd like to be able to generate then perform symbolic calculations with the family of polynomials defined for all $n\in \mathbb{N}$ and all $k=0,\dotsc, 2n$ by $p_{n,k}=\begin{cases} 0 & \textrm{if } k=0\newline \sum_{j=1}^k x_j&\textrm{if } k\leq n\newline \sum_{j=1}^{2n-k+1}x_j&\textrm{if }k>n \end{cases}$

So far the attempts that I've had are of the form:

sage: h = lambda k:sum([var('d_%d' %(i+1)) for i in range(k)])

but I don't seem to easily perform calculations with these. Another method I was trying is defining $\mathbb{Q}[x_0,\dotsc, x_n]$ then trying to define these polynomials using conditional statements. I seem to keep getting errors stating my variables don't exist.

Would love some help or a hint.

Comments

Your $p_k$ silently depends on $n$ . So you want to fix $n$ and study the $p_k$ ?

vdelecroix ( 6 years ago )

As you write them, your $p_k$ are first-degrees polynomials of $n$ variables at most. Is that really what you want ?

Emmanuel Charpentier ( 6 years ago )

That is correct, I'd like to fix any $n$ then study the $p_k$ .

Yes, these will be degree $1$ polynomials in at most $n$ variables.

DCL ( 6 years ago )

Comments:

it might make sense to call the polynomials $p_{n, k}$ .
in the case $0 < k \le n$ , the formula should have $x_j$ , not $x_k$
indenting the code line h = lambda k: ... by 4 spaces will make it display as code

slelievre ( 6 years ago )

Thanks for taking all the comments into account and editing your question!

slelievre ( 6 years ago )

add a comment

def p(n, k): if n not in NN or k not in NN or k > 2*n: raise ValueError("Unsupported values: n = {}, k = {}".format(n, k)) if k == 0: return R.zero() if 1 <= k <= n: return sum(x[j] for j in (1 .. k)) # case n < k <= 2* n: return sum(x[j] for j in (1 .. 2*n-k+1))

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Defining family of multivariable polynomials

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