# Multivariate symbolics

Consider this piece of code:

sage: n=4    # dimension of state-space
sage: x = polygens(QQ, ['x'+str(i) for i in [1..n]]); x    # vector of state variables
(x1, x2, x3, x4)


There is a handy keyword with formal power series, the number of generators (num_gens), and it works like:

sage: PowerSeriesRing(QQ, 'x', num_gens=5)
Multivariate Power Series Ring in x0, x1, x2, x3, x4 over Rational Field


There is a kind of analogue with polynomials, but not keyworded:

sage: PolynomialRing(QQ, 'x', 5)
Multivariate Polynomial Ring in x0, x1, x2, x3, x4 over Rational Field


Is it possible to use any of these to define a vector of polynomial variables in the object x as the beginning (but without list comprehension)?

As a side note, i think the var constructor also lacks a vector form?

sage: var(['x'+str(i) for i in [1..n]])
(x1, x2, x3, x4)


This could be useful to simplify constructions as:

sage: A = matrix(SR, [['a'+str(i)+str(j) for j in [1..n]] for i in [1..n]]); A

[a11 a12 a13 a14]
[a21 a22 a23 a24]
[a31 a32 a33 a34]
[a41 a42 a43 a44]


In Matlab this is A = sym('a', n), but i didn't find a similar shortcut in Sage.

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sage: PolynomialRing(QQ, 'x', 5).gens()
(x0, x1, x2, x3, x4)


sage: matrix(SR,4,4, lambda i,j: SR.symbol("a%s%s"%(i,j)))
[a00 a01 a02 a03]
[a10 a11 a12 a13]
[a20 a21 a22 a23]
[a30 a31 a32 a33]

more

yes, indeed gens answers my question! i had a closer look at polygens code, and seemingly a trivial change allows to do polygens(QQ, 'x', 4). this is #22809

for the symbolic vector/matrix, i would take your solution and embed it into calculus.var so that var('x', 4) and var('x', 4, 4) work and produce a vector/matrix respectively .. what do you think?

var(*args) already means the same thing as for a in args: var(a), so the signature is not available for different interpretation.