This CD holds a collection of for some operations of polynomials over rings.
The data structures for polynomials can be arithmetic expressions, for
instance using the ring1.expression symbol, or DMP as
in the CD polyd1.
This symbol is a unary function, whose argument should be a polynomial f.
When applied to f, it represents a list of factors of f.
Cf. polynomial4.factorisations for a description of the expected reply.
This symbol may be used in the reply of polynomial4.factorise and
takes at least 1 argument.
The first argument is one of polynomial4.factorisations_complete to
indicate that the following list of polynomial4.factors cells covers
all possible factorisations. The counterpart would be
polynomial4.factorisations_possibly_incomplete to indicate that the
following list of factorisations are some of possibly many more
factorisations.
Note: If the polynomial algebra is a UFD (unique factorisation domain)
the uniqueness can be underpinned by giving exactly one
polynomial4.factors cell and using the symbol
polynomial4.factorisations_complete here.
The rest of the arguments are polynomial4.factors cells, each of
which being a possible factorisation.
Using the call of polynomial4.factorise above we might obtain:
Example:
<OMOBJ xmlns="http://www.openmath.org/OpenMath">
<OMA>
<OMS name="factorisations" cd="polynomial4"/>
<!-- are there more to come? -->
<OMS name="factorisations_complete" cd="polynomial4"/>
<!-- now the rain dance -->
<OMA>
<OMS name="factors" cd="polynomial4"/>
<!-- 1st arg, indicate how certain we are about this -->
<OMS name="definitely_irreducible" cd="polynomial4"/>
<!-- 2nd arg, we did the flip over the following domain -->
<OMA>
<OMS name="poly_ring_d_named" cd="polyd1"/>
<OMS name="Z" cd="ringname1"/>
<OMV name="X"/>
</OMA>
<!-- the common coefficient -->
<OMA>
<OMS name="one" cd="alg1"/>
<OMS name="Z" cd="ringname1"/>
</OMA>
<!-- the 1st factor -->
<OMA>
<OMS name="factor" cd="polynomial4"/>
<OMA>
<OMS name="DMP" cd="polyd1"/>
<OMA>
<OMS name="poly_ring_d_named" cd="polyd1"/>
<OMS name="Z" cd="ringname1"/>
<OMV name="X"/>
</OMA>
<OMA>
<OMS name="SDMP" cd="polyd1"/>
<OMA>
<OMS name="term" cd="polyd1"/>
<OMI> 1 </OMI>
<OMI> 1 </OMI>
</OMA>
<OMA>
<OMS name="term" cd="polyd1"/>
<OMI> -1 </OMI>
<OMI> 0 </OMI>
</OMA>
</OMA>
</OMA>
<OMA>
<OMS name="multiplicity" cd="polynomial4"/>
<OMI> 1 </OMI>
</OMA>
</OMA>
<!-- the 2nd factor -->
<OMA>
<OMS name="factor" cd="polynomial4"/>
<OMA>
<OMS name="DMP" cd="polyd1"/>
<OMA>
<OMS name="poly_ring_d_named" cd="polyd1"/>
<OMS name="Z" cd="ringname1"/>
<OMV name="X"/>
</OMA>
<OMA>
<OMS name="SDMP" cd="polyd1"/>
<OMA>
<OMS name="term" cd="polyd1"/>
<OMI> 1 </OMI>
<OMI> 1 </OMI>
</OMA>
<OMA>
<OMS name="term" cd="polyd1"/>
<OMI> 1 </OMI>
<OMI> 0 </OMI>
</OMA>
</OMA>
</OMA>
<OMA>
<OMS name="multiplicity" cd="polynomial4"/>
<OMI> 1 </OMI>
</OMA>
</OMA>
</OMA>
</OMA>
</OMOBJ>
This symbol is used in the reply of polynomial4.factorise and takes
at least 2 arguments. Note this symbol may also be used in a
polynomial4.factorisations cell.
The first argument is one of polynomial4.definitely_irreducible or
polynomial4.possibly_reducible and specifies whether the computed
factorisation is known to be irreducible or if the irreducibility of some
of the factors is not guaranteed.
Note: This symbol is mandatory even if the factors themselves (see
polynomial4.factor) can carry that information, this is simply to
connive at computer algebra systems that cannot figure out which of the
factors is the possibly reducible one. Generally this slot must be
polynomial4.possibly_reducible if at least one of the factors is
possibly reducible.
The second argument contains a polyd1.poly_ring_d or
polyd1.poly_ring_d_named cell, as specified in e.g. polyd
or polyd1 to indicate the underlying polynomial algebra.
The third argument is a symbol polynomial4.common_coefficient and
denotes the common coefficient of the factorisation.
Note: In case the ground ring itself is regarded as being injected into
the polynomial algebra, or the factorisation is normalised, this field may
be used to specify the unit giving the normalisation.
Furthermore, the cell comprises polynomial4.factor cells which in turn
represent the factors of the polynomial in a factorisation along with
their multiplicities.
Using the call of polynomial4.factorise above we might obtain:
A symbol which represents one factor of a factorisation, it takes at least
2 arguments, the first of which being the factor polynomial, e.g. a
polyd1.DMP, and the second being its multiplicity
specified as an integer >= 1.
Optionally, the third argument is one of
polynomial4.definitely_irreducible,
polynomial4.possibly_reducible to indicate whether or not the given
factor is guaranteed to be irreducible.
Furthermore, this symbol may contain polynomial4.ground_ring_injected
to indicate that the ground ring is considered to be embedded in the
polynomial algebra and hence the factor is actually the factorisation of
a polynomial coefficient.
A symbol which denotes that the ground ring of a polynomial algebra is
considered to be part of the latter. This is used in the
polynomial4.factor symbol to indicate that the factor is part of the
factorisation of the common coefficient.
This symbol is a binary function whose arguments are polynomials f and g
which must be defined over the same ground domain.
When applied to f and g, it represents the quotient arising from dividing
f by g and the remainder h such that h is congruent f modulo g.
The result is gathered in a polynomial4.quotient_remainder cell.
Hint: We consider named polynomial rings, i.e. the indeterminate is
explicitly specified by a named variable, different once the variable
names differ. That is, a polynomial in Z[X] cannot be divided by a
polynomial in Z[Y] a priori. However, we leave it up to the implementor
to handle this differently, though we strongly encourage implementors
to return a polynomial in an anonymous indeterminate (using
e.g. polyd1.poly_ring_d rather than polyd1.poly_ring_d_named).
This symbol is a container for the result of polynomial4.divide.
It takes 2 arguments in unspecified order, polynomial4.quotient
and polynomial4.remainder.
Using the above polynomial4.divide call we may obtain:
