Build a polynomial from it's evaluation : genpoly

genpoly takes three arguments : a polynomial P with n - 1 variables, an integer b and the name of a variable var.
genpoly returns the polynomial Q with n variables (the P variables and the variable var given as second argument), such that :

• subst(Q,var=b)==P
• the coefficients of Q belongs to the interval ] - b/2 ; b/2]

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). Input :

genpoly(61,6,x)

Output :

2*x^2-2*x+1

Indeed 61 divided by 6 is 10, remains 1, then 10 divided by 6 is 2 remains -2 (instead of the usual quotient 1 and remainder 4 out of bounds),

61 = 2*6² - 2*6 + 1

Input :

genpoly(5,6,x)

Output :

x-1

Indeed : 5 = 6 - 1
Input :

genpoly(7,6,x)

Output :

x+1

Indeed : 7 = 6 + 1
Input :

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

Output :

x*y+x+y-1

Indeed : x*y + x + y - 1 = y(x + 1) + (x - 1)
Input :

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

Output :

x*y+x*z+y-z

Indeed : x*y + x*z + y - z = y*(x + 1) + z*(x - 1)