uinv_as_power_series, ureverse_inv_as_power_series

uinv_as_power_series(p,d)

ureverse_inv_as_power_series(p,d)

:: Computes the truncated inverse as a power series.

return

univariate polynomial

p

univariate polynomial

d

non-negative integer

• For a polynomial p with a non zero constant term, uinv_as_power_series(p,d) computes a polynomial r whose degree is at most d such that p*r = 1 mod x^(d+1), where x is the variable of p.
• Let e be the degree of p. ureverse_inv_as_power_series(p,d) computes uinv_as_power_series(p1,d) for p1=ureverse(p,e).
• The output of ureverse_inv_as_power_series() can be used as the input of rembymul_precomp().

[123] A=(x+1)^5;                 
x^5+5*x^4+10*x^3+10*x^2+5*x+1
[124] uinv_as_power_series(A,5); 
-126*x^5+70*x^4-35*x^3+15*x^2-5*x+1
[126] A*R;
-126*x^10-560*x^9-945*x^8-720*x^7-210*x^6+1
[127] A=x^10+x^9;
x^10+x^9
[128] R=ureverse_inv_as_power_series(A,5);
-x^5+x^4-x^3+x^2-x+1
[129] ureverse(A)*R;
-x^6+1

References

section utrunc, udecomp, ureverse, section udiv, urem, urembymul, urembymul_precomp, ugcd.

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