CoCoA:QZP
QZP
change field for polynomials and ideals
Description
The functions QZP and <ttref>ZPQ</ttref> map polynomials and
ideals of other rings into ones of the current ring.
When mapping from one ring to another, one of the rings must have coefficients in the rational numbers and the other must have coefficients in a finite field. The indeterminates in both rings must be identical.
The function QZP maps polynomials with rational coefficients to
polynomials with coefficients in a finite field; the function <ttref>ZPQ</ttref>
does the reverse, mapping a polynomial with finite field coefficients into one with rational (actually, integer) coefficients. The function <ttref>ZPQ</ttref> is not uniquely defined mathematically, and currently for each coefficient the least non-negative equivalent integer is chosen. Users should not rely on this choice, though any change will be documented.
Example
Use R ::= Q[x,y,z];
F := 1/2*x^3 + 34/567*x*y*z - 890; -- a poly with rational coefficients
Use S ::= Z/(101)[x,y,z];
QZP(F); -- compute its image with coeffs in Z/(101)
-50x^3 - 19xyz + 19
-------------------------------
G := It;
Use R;
ZPQ(G); -- now map that result back to Q[x,y,z]
-- it is NOT the same as F...
51x^3 + 82xyz + 19
-------------------------------
H := It;
F - H; -- ... but the difference is divisible by 101
-101/2x^3 - 46460/567xyz - 909
-------------------------------
Use S;
QZP(H) - G; -- F and H have the same image in Z/(101)[x,y,z]
0
-------------------------------
Syntax
QZP(F:POLY):POLY QZP(F:LIST of POLY):LIST of POLY QZP(I:IDEAL):IDEAL
Ring Mappings: the Image Function
<type>ideal</type> <type>polynomial</type> <type>ring</type>
