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Content

Content of polynomials over algebraic extension fields

Primpart

Primitive part of polynomials over algebraic extension fields

 Calling Sequence evala(Content(P, x, 'pp'), opts) evala(Primpart(P, x, 'co'), opts)

Parameters

 P - multivariate polynomial in x over an algebraic extension field. x - (optional) name or set or list of names. pp - (optional) unevaluated name. co - (optional) unevaluated name. opts - (optional) an option name or a set of option names. - Options currently supported: independent.

Description

 • These functions compute contents and primitive parts of multivariate polynomials with coefficients in algebraic number and function fields.
 • Algebraic functions and algebraic numbers may be represented by radicals or with the RootOf notation (see type,algnum, type,algfun, type,radnum, type,radfun).
 • Generally speaking, if P is a polynomial in R[x] where R is a GCD domain, then a content co of P is a greatest common divisor of the coefficients of P. The associated primitive part is the polynomial pp of R[x] such that P(x) = co * pp(x). The content and the primitive part are unique up to the choice a unit of R.
 The above definition can be generalized to polynomials with coefficients in the fraction field of R by clearing out the common denominator of the coefficients of P and assigning it to the content. In this case, the content is in Frac(R).
 • Since there is no way in Maple to specify R, Content and Primpart attempt to make a reasonable guess and proceed accordingly: R will typically be Z -- the ring of integers, a polynomial ring over the integers or a polynomial ring over an algebraic number or function field.
 • More precisely, if the coefficients of P are polynomials or rational functions over the rational numbers, Content and Primpart return the same result as content and primpart.
 If P is a polynomial in Q(A)(y)[x], where:
 – Q denotes the set of rational numbers,
 – A denotes a (non-empty) set of algebraic numbers and algebraic functions,
 – y is a set of names in P which are not algebraic indeterminates, that is, which do not appear inside a RootOf or a radical,
 – x is a set of indeterminates,
 then, a primitive part of P is a polynomial pp in Q(A)[y][x] such that the coefficients of pp in x are relatively prime polynomials of Q(A)[y]. The associated content is an element co of Q(A)(y) such that P(x) = co * pp(x). In this case, a primitive part and its corresponding content are defined up to a multiplicative constant in Q(A). In particular, Content and Primpart may not use the same normalization coefficient. In other words, P may not be the product of the results of Content and Primpart. To obtain the associated factors, use the optional arguments (see below).
 The function Primpart returns a primitive part whose denominator is 1. The result, viewed formally as a polynomial Z[A,y,x] -- where the algebraic quantities in A are frozen -- is also primitive. Note that the value assigned to the optional argument of Content may not satisfy this property (see below).
 The function Content returns a content normalized in the following way: the leading coefficient of the numerator and of the denominator are rationalized -- i.e. they do not contain any algebraic quantities. Note that the value assigned to the optional argument of Primpart may not satisfy this property (see below).
 Since the ordering of variables may vary from a session to another, the leading coefficients may change.
 • Algebraic numbers and functions occurring in the results are reduced modulo their minimal polynomial (see Normal).
 • If the optional argument pp of Content is given, then it is assigned the primitive part associated to the content.
 • If the optional argument co of Primpart is given, then it is assigned the content associated to the primitive part.
 • If x is not specified, then Primpart and Content will use all the names in P which are not algebraic indeterminates (see above).
 • The functions Primpart and Content preserve partial factorization as far possible: The output may not be expanded.
 • The RootOf and the radicals defining the algebraic numbers must form an independent set of algebraic quantities, otherwise an error is returned. Note that this condition needs not be satisfied if the expression contains only algebraic numbers in radical notation (i.e. 2^(1/2), 3^(1/2), 6^(1/2)). For, a basis over Q for the radicals can be computed by Maple in this case.
 • To skip the independence checking, use the option independent.
 • If P contains functions, their arguments are normalized and the functions are frozen before the computation proceeds.
 • Other objects are frozen and considered as variables.

Examples

 > $f≔\frac{\mathrm{sqrt}\left(t\right)y+1}{2y-2}$
 ${f}{≔}\frac{\sqrt{{t}}{}{y}{+}{1}}{{2}{}{y}{-}{2}}$ (1)
 > $a≔\mathrm{expand}\left(f\left(\mathrm{sqrt}\left(t\right)+x\right)\right)$
 ${a}{≔}\frac{{t}{}{y}}{{2}{}{y}{-}{2}}{+}\frac{\sqrt{{t}}{}{y}{}{x}}{{2}{}{y}{-}{2}}{+}\frac{\sqrt{{t}}}{{2}{}{y}{-}{2}}{+}\frac{{x}}{{2}{}{y}{-}{2}}$ (2)
 > $\mathrm{co1}≔\mathrm{evala}\left(\mathrm{Content}\left(a,x,'\mathrm{pp1}'\right)\right)$
 ${\mathrm{co1}}{≔}\frac{{t}{}{y}{+}\sqrt{{t}}}{{y}{-}{1}}$ (3)
 > $\mathrm{pp1}$
 $\frac{\sqrt{{t}}{}{x}{+}{t}}{{2}{}{t}}$ (4)
 > $\mathrm{evala}\left(\mathrm{Normal}\left(a-\mathrm{co1}\mathrm{pp1}\right)\right)$
 ${0}$ (5)
 > $\mathrm{pp2}≔\mathrm{evala}\left(\mathrm{Primpart}\left(a,x,'\mathrm{co2}'\right)\right)$
 ${\mathrm{pp2}}{≔}\sqrt{{t}}{}{x}{+}{t}$ (6)
 > $\mathrm{co2}$
 $\frac{{t}{}{y}{+}\sqrt{{t}}}{{2}{}{t}{}\left({y}{-}{1}\right)}$ (7)
 > $\mathrm{evala}\left(\mathrm{Normal}\left(a-\mathrm{co2}\mathrm{pp2}\right)\right)$
 ${0}$ (8)
 > $\mathrm{evala}\left(\mathrm{Normal}\left(\frac{\mathrm{co1}}{\mathrm{co2}}\right)\right)$
 ${2}{}{t}$ (9)
 > $\mathrm{evala}\left(\mathrm{Content}\left({x}^{2}-2+\left(x-\mathrm{RootOf}\left({x}^{2}-2\right)\right)y,y\right)\right)$
 ${x}{-}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{-}{2}\right)$ (10)

If a polynomial defining a RootOf is reducible, the RootOf does not generate a well-defined field. An error is returned:

 > $\mathrm{evala}\left(\mathrm{Content}\left({x}^{2}y+\left(x-\mathrm{RootOf}\left({x}^{2}-x\right)\right),y\right)\right)$

To pretend that all the defining polynomials are irreducible, use the option 'independent':

 > $\mathrm{evala}\left(\mathrm{Content}\left({x}^{2}y+\left(x-\mathrm{RootOf}\left({x}^{2}-x\right)\right),y\right),'\mathrm{independent}'\right)$
 ${1}$ (11)

Alternatively, use indexed RootOfs:

 > $\mathrm{evala}\left(\mathrm{Content}\left({x}^{2}y+\left(x-\mathrm{RootOf}\left({x}^{2}-x,\mathrm{index}=1\right)\right),y\right)\right)$
 ${x}$ (12)
 > $\mathrm{evala}\left(\mathrm{Content}\left({x}^{2}y+\left(x-\mathrm{RootOf}\left({x}^{2}-x,\mathrm{index}=2\right)\right),y\right)\right)$
 ${1}$ (13)