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B.1 Representation of mathematical objects

SINGULAR distinguishes between objects which do not belong to a ring and those which belong to a specific ring (see Rings and orderings). We comment only on the latter ones.

Internally all ring-dependent objects are polynomials or structures built from polynomials (and some additional information). Note that SINGULAR stores (and hence prints) a polynomial automatically w.r.t. the monomial ordering.

The definition of ideals and matrices, respectively, is straight forward: The user gives a list of polynomials which generate the ideal, resp. which are the entries of the matrix. (The number of rows and columns need to be provided when creating the matrix.)

A vector in SINGULAR is always an element of a free module over the basering. It is given as a list of polynomials in one of the following formats $[f_1,...,f_n]$ or $f_1*gen(1)+...+f_n*gen(n)$, where $gen(i)$denotes the i-th canonical generator of a free module (with 1 at index i and 0 everywhere else). Both forms are equivalent. A vector is internally represented in the second form with the $gen(i)$being "special" ring variables, ordered accordingly to the monomial ordering. Therefore, the form $[f_1,...,f_n]$serves as output only if the monomial ordering gives priority to the component, i.e., is of the form (c,...) (see Module orderings). However, in any case the procedure show from the library inout.lib displays the bracket format.

A vector $v=[f_1,...,f_n]$should always be considered as a column vector in a free module of rank equal to nrows($v$)where nrows($v$)is equal to the maximal index $r$such that $f_r \not= 0$.This is due to the fact, that internally $v$is a polynomial in a sparse representation, i.e., $f_i*gen(i)$is not stored if $f_i=0$(for reasons of efficiency), hence the last 0-entries of $v$are lost. Only more complex structures are able to keep the rank.

A module $M$in SINGULAR is given by a list of vectors $v_1,...,v_k$which generate the module as a submodule of the free module of rank equal to nrows($M$)which is the maximum of nrows($v_i$).

If one wants to create a module with a larger rank than given by its generators, one has to use the command attrib(M,"rank",r) (see attrib, nrows) or to define a matrix first, then converting it into a module. Modules in SINGULAR are almost the same as matrices, they may be considered as sparse representations of matrices. A module of a matrix is generated by the columns of the matrix and a matrix of a module has as columns the generators of the module. These conversions preserve the rank and the number of generators, resp. the number of rows and columns.

By the above remarks it might appear that SINGULAR is only able to handle submodules of a free module. However, this is not true. SINGULAR can compute with any finitely generated module over the basering $R$.Such a module, say $N$,is not represented by its generators but by its (generators and) relations. This means that $N = R^n/M$ where $n$is the number of generators of $N$ and $M \subseteq R^n$is the module of relations. In other words, defining a module $M$as a submodule of a free module $R^n$can also be considered as the definition of $N = R^n/M$.

Note that most functions, when applied to a module $M$,really deal with $M$.However, there are some functions which deal with $N = R^n/M$ instead of $M$.

For example, std(M) computes a standard basis of $M$(and thus gives another representation of $N$ as $N = R^n/$std($M$)).However, dim(M), resp. vdim(M), return dim$(R^n/M)$, resp. dim$_k(R^n/M)$(if M is given by a standard basis).

The function syz(M) returns the first syzygy module of $M$,i.e., the module of relations of the given generators of $M$which is equal to the second syzygy module of $N$.Refer to the description of each function in Functions to get information which module the function deals with.

The numbering in res and other commands for computing resolutions refers to a resolution of $N = R^n/M$(see res; Syzygies and resolutions).

It is possible to compute in any field which is a valid ground field in SINGULAR. For doing so, one has to define a ring with the desired ground field and at least one variable. The elements of the field are of type number, but may also be considered as polynomials (of degree 0). Large computations should be faster if the elements of the field are defined as numbers.

The above remarks do also apply to quotient rings. Polynomial data are stored internally in the same manner, the only difference is that this polynomial representation is in general not unique. reduce(f,std(0)) computes a normal form of a polynomial f in a quotient ring (cf. reduce).

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