Building a polynomial from its evaluation

11.5.4 Building a polynomial from its evaluation

The genpoly command finds a polynomial which evaluates to a given polynomial.

genpoly takes three arguments:
- P, a polynomial with n−1 variables.
- b, an integer.
- x, the name of a variable.
genpoly(P,b,x) returns the polynomial Q with n variables (the n−1 variables in P and the variable x) such that the coefficients of Q are in the interval (−b/2,b/2] and Q|_x=b= P. In other words, P is written in base b but using the convention that the Euclidean remainder belongs to (−b/2,b/2] (this convention is also known as s-mod representation).

Examples

genpoly(61,6,x)

2 x²−2 x+1

Indeed 61 divided by 6 is 10 with remainder 1, then 10 divided by 6 is 2 with remainder −2 (instead of the usual quotient 1 and remainder 4 out of bounds), 61=2· 6²−2· 6+1.

genpoly(5,6,x)

x−1

Indeed, 5=6−1.

genpoly(7,6,x)

x+1

Indeed, 7=6+1.

genpoly(7*y+5,6,x)

x y+x+y−1

Indeed, xy+x+y−1=y(x+1)+(x−1).

genpoly(7*y+5*z^2,6,x)

x y+x z²+y−z²

Indeed, xy+xz+y−z=y(x+1)+z(x−1).