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QZP --
change field for polynomials and ideals
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QZP(F: RINGELEM): RINGELEM
QZP(F: LIST of POLY): LIST of POLY
QZP(I: IDEAL): IDEAL |
***** NOT YET IMPLEMENTED *****
See example below
The functions
QZP and
ZPQ
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
ZPQ
does the reverse, mapping a polynomial with finite field coefficients
into one with rational (actually, integer) coefficients. The function
ZPQ
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.
/**/ use R ::= QQ[x,y,z];
/**/ F := 1/2*x^3 + (34/567)*x*y*z - 890; -- a poly with rational coefficients
/**/ use S ::= ZZ/(101)[x,y,z];
/**/ -- this is the clean way to do it!
/**/ phi := PolyRingHom(R, S, QQEmbeddingHom(S), indets(S));
/**/ phi(F);
-50*x^3 -19*x*y*z +19
***** NOT YET IMPLEMENTED *****
QZP(F); -- compute its image with coeffs in ZZ/(101)
-50x^3 - 19xyz + 19
-------------------------------
G := It;
use R;
ZPQ(G); -- now map that result back to QQ[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 ZZ/(101)[x,y,z]
0
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