Reducing an expressing modulo a variable expression

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For a symbolic approach you can use a substitution with a wildcard:

sage: var('p,a')
sage: f = (a*p+1)^2
sage: w0 = SR.wild()
sage: f.expand().subs({p^w0 : 0})
2*a*p + 1

This makes use of the fact that p^1 does not appear in an expanded symbolic expression.

Using a different ring would probably be more efficient, though.

If you matrices have polynomial entries, a possible approach is to define a polynomial ring in all variables but p, and the power series ring in p over that ring like:

R.<a11,a12,a13> = PolynomialRing(QQ)
P.<p> = PowerSeriesRing(R,default_prec=2)

Then to ignore the powers p^2 and higher, one can add O(p^2} and convert the result to polynomial if needed:

f = a11 + a12*p + a13*p^2
f += O(p^2)
print(f)
print(f.polynomial())