# Free associative unital algebras, implemented via Singular’s letterplace rings¶

AUTHOR:

With this implementation, Groebner bases out to a degree bound and normal forms can be computed for twosided weighted homogeneous ideals of free algebras. For now, all computations are restricted to weighted homogeneous elements, i.e., other elements can not be created by arithmetic operations.

EXAMPLES:

sage: F.<x,y,z> = FreeAlgebra(QQ, implementation='letterplace')
sage: F
Free Associative Unital Algebra on 3 generators (x, y, z) over Rational Field
sage: I = F*[x*y+y*z,x^2+x*y-y*x-y^2]*F
sage: I
Twosided Ideal (x*y + y*z, x*x + x*y - y*x - y*y) of Free Associative Unital Algebra on 3 generators (x, y, z) over Rational Field
sage: x*(x*I.0-I.1*y+I.0*y)-I.1*y*z
x*y*x*y + x*y*y*y - x*y*y*z + x*y*z*y + y*x*y*z + y*y*y*z
sage: x^2*I.0-x*I.1*y+x*I.0*y-I.1*y*z in I
True


The preceding containment test is based on the computation of Groebner bases with degree bound:

sage: I.groebner_basis(degbound=4)
Twosided Ideal (y*z*y*y - y*z*y*z + y*z*z*y - y*z*z*z, y*z*y*x + y*z*y*z + y*z*z*x + y*z*z*z, y*y*z*y - y*y*z*z + y*z*z*y - y*z*z*z, y*y*z*x + y*y*z*z + y*z*z*x + y*z*z*z, y*y*y - y*y*z + y*z*y - y*z*z, y*y*x + y*y*z + y*z*x + y*z*z, x*y + y*z, x*x - y*x - y*y - y*z) of Free Associative Unital Algebra on 3 generators (x, y, z) over Rational Field


When reducing an element by $$I$$, the original generators are chosen:

sage: (y*z*y*y).reduce(I)
y*z*y*y


However, there is a method for computing the normal form of an element, which is the same as reduction by the Groebner basis out to the degree of that element:

sage: (y*z*y*y).normal_form(I)
y*z*y*z - y*z*z*y + y*z*z*z
sage: (y*z*y*y).reduce(I.groebner_basis(4))
y*z*y*z - y*z*z*y + y*z*z*z


The default term order derives from the degree reverse lexicographic order on the commutative version of the free algebra:

sage: F.commutative_ring().term_order()
Degree reverse lexicographic term order


A different term order can be chosen, and of course may yield a different normal form:

sage: L.<a,b,c> = FreeAlgebra(QQ, implementation='letterplace', order='lex')
sage: L.commutative_ring().term_order()
Lexicographic term order
sage: J = L*[a*b+b*c,a^2+a*b-b*c-c^2]*L
sage: J.groebner_basis(4)
Twosided Ideal (2*b*c*b - b*c*c + c*c*b, a*c*c - 2*b*c*a - 2*b*c*c - c*c*a, a*b + b*c, a*a - 2*b*c - c*c) of Free Associative Unital Algebra on 3 generators (a, b, c) over Rational Field
sage: (b*c*b*b).normal_form(J)
1/2*b*c*c*b - 1/2*c*c*b*b


Here is an example with degree weights:

sage: F.<x,y,z> = FreeAlgebra(QQ, implementation='letterplace', degrees=[1,2,3])
sage: (x*y+z).degree()
3


Todo

The computation of Groebner bases only works for global term orderings, and all elements must be weighted homogeneous with respect to positive integral degree weights. It is ongoing work in Singular to lift these restrictions.

We support coercion from the letterplace wrapper to the corresponding generic implementation of a free algebra (FreeAlgebra_generic), but there is no coercion in the opposite direction, since the generic implementation also comprises non-homogeneous elements.

We also do not support coercion from a subalgebra, or between free algebras with different term orderings, yet.

class sage.algebras.letterplace.free_algebra_letterplace.FreeAlgebra_letterplace

Finitely generated free algebra, with arithmetic restricted to weighted homogeneous elements.

NOTE:

The restriction to weighted homogeneous elements should be lifted as soon as the restriction to homogeneous elements is lifted in Singular’s “Letterplace algebras”.

EXAMPLES:

sage: K.<z> = GF(25)
sage: F.<a,b,c> = FreeAlgebra(K, implementation='letterplace')
sage: F
Free Associative Unital Algebra on 3 generators (a, b, c) over Finite Field in z of size 5^2
sage: P = F.commutative_ring()
sage: P
Multivariate Polynomial Ring in a, b, c over Finite Field in z of size 5^2


We can do arithmetic as usual, as long as we stay (weighted) homogeneous:

sage: (z*a+(z+1)*b+2*c)^2
(z + 3)*a*a + (2*z + 3)*a*b + (2*z)*a*c + (2*z + 3)*b*a + (3*z + 4)*b*b + (2*z + 2)*b*c + (2*z)*c*a + (2*z + 2)*c*b - c*c
sage: a+1
Traceback (most recent call last):
...
ArithmeticError: Can only add elements of the same weighted degree

commutative_ring()

Return the commutative version of this free algebra.

NOTE:

This commutative ring is used as a unique key of the free algebra.

EXAMPLES:

sage: K.<z> = GF(25)
sage: F.<a,b,c> = FreeAlgebra(K, implementation='letterplace')
sage: F
Free Associative Unital Algebra on 3 generators (a, b, c) over Finite Field in z of size 5^2
sage: F.commutative_ring()
Multivariate Polynomial Ring in a, b, c over Finite Field in z of size 5^2
sage: FreeAlgebra(F.commutative_ring()) is F
True

current_ring()

Return the commutative ring that is used to emulate the non-commutative multiplication out to the current degree.

EXAMPLES:

sage: F.<a,b,c> = FreeAlgebra(QQ, implementation='letterplace')
sage: F.current_ring()
Multivariate Polynomial Ring in a, b, c over Rational Field
sage: a*b
a*b
sage: F.current_ring()
Multivariate Polynomial Ring in a, b, c, a_1, b_1, c_1 over Rational Field
sage: F.set_degbound(3)
sage: F.current_ring()
Multivariate Polynomial Ring in a, b, c, a_1, b_1, c_1, a_2, b_2, c_2 over Rational Field

degbound()

Return the degree bound that is currently used.

NOTE:

When multiplying two elements of this free algebra, the degree bound will be dynamically adapted. It can also be set by set_degbound().

EXAMPLES:

In order to avoid we get a free algebras from the cache that was created in another doctest and has a different degree bound, we choose a base ring that does not appear in other tests:

sage: F.<x,y,z> = FreeAlgebra(ZZ, implementation='letterplace')
sage: F.degbound()
1
sage: x*y
x*y
sage: F.degbound()
2
sage: F.set_degbound(4)
sage: F.degbound()
4

gen(i)

Return the $$i$$-th generator.

INPUT:

$$i$$ – an integer.

OUTPUT:

Generator number $$i$$.

EXAMPLES:

sage: F.<a,b,c> = FreeAlgebra(QQ, implementation='letterplace')
sage: F.1 is F.1  # indirect doctest
True
sage: F.gen(2)
c

generator_degrees()
ideal_monoid()

Return the monoid of ideals of this free algebra.

EXAMPLES:

sage: F.<x,y> = FreeAlgebra(GF(2), implementation='letterplace')
sage: F.ideal_monoid()
Monoid of ideals of Free Associative Unital Algebra on 2 generators (x, y) over Finite Field of size 2
sage: F.ideal_monoid() is F.ideal_monoid()
True

is_commutative()

Tell whether this algebra is commutative, i.e., whether the generator number is one.

EXAMPLES:

sage: F.<x,y,z> = FreeAlgebra(QQ, implementation='letterplace')
sage: F.is_commutative()
False
sage: FreeAlgebra(QQ, implementation='letterplace', names=['x']).is_commutative()
True

is_field(proof=True)

Tell whether this free algebra is a field.

NOTE:

This would only be the case in the degenerate case of no generators. But such an example can not be constructed in this implementation.

ngens()

Return the number of generators.

EXAMPLES:

sage: F.<a,b,c> = FreeAlgebra(QQ, implementation='letterplace')
sage: F.ngens()
3

set_degbound(d)

Increase the degree bound that is currently in place.

NOTE:

The degree bound can not be decreased.

EXAMPLES:

In order to avoid we get a free algebras from the cache that was created in another doctest and has a different degree bound, we choose a base ring that does not appear in other tests:

sage: F.<x,y,z> = FreeAlgebra(GF(251), implementation='letterplace')
sage: F.degbound()
1
sage: x*y
x*y
sage: F.degbound()
2
sage: F.set_degbound(4)
sage: F.degbound()
4
sage: F.set_degbound(2)
sage: F.degbound()
4

term_order_of_block()

Return the term order that is used for the commutative version of this free algebra.

EXAMPLES:

sage: F.<x,y,z> = FreeAlgebra(QQ, implementation='letterplace')
sage: F.term_order_of_block()
Degree reverse lexicographic term order
sage: L.<a,b,c> = FreeAlgebra(QQ, implementation='letterplace',order='lex')
sage: L.term_order_of_block()
Lexicographic term order

sage.algebras.letterplace.free_algebra_letterplace.poly_reduce(ring=None, interruptible=True, attributes=None, *args)

This function is an automatically generated C wrapper around the Singular function ‘NF’.

This wrapper takes care of converting Sage datatypes to Singular datatypes and vice versa. In addition to whatever parameters the underlying Singular function accepts when called, this function also accepts the following keyword parameters:

INPUT:

• args – a list of arguments
• ring – a multivariate polynomial ring
• interruptible – if True pressing Ctrl-C during the execution of this function will interrupt the computation (default: True)
• attributes – a dictionary of optional Singular attributes assigned to Singular objects (default: None)

If ring is not specified, it is guessed from the given arguments. If this is not possible, then a dummy ring, univariate polynomial ring over QQ, is used.

EXAMPLES:

sage: groebner = sage.libs.singular.function_factory.ff.groebner
sage: P.<x, y> = PolynomialRing(QQ)
sage: I = P.ideal(x^2-y, y+x)
sage: groebner(I)
[x + y, y^2 - y]
sage: triangL = sage.libs.singular.function_factory.ff.triang__lib.triangL
sage: P.<x1, x2> = PolynomialRing(QQ, order='lex')
sage: f1 = 1/2*((x1^2 + 2*x1 - 4)*x2^2 + 2*(x1^2 + x1)*x2 + x1^2)
sage: f2 = 1/2*((x1^2 + 2*x1 + 1)*x2^2 + 2*(x1^2 + x1)*x2 - 4*x1^2)
sage: I = Ideal(Ideal(f1,f2).groebner_basis()[::-1])
sage: triangL(I, attributes={I:{'isSB':1}})
[[x2^4 + 4*x2^3 - 6*x2^2 - 20*x2 + 5, 8*x1 - x2^3 + x2^2 + 13*x2 - 5],
[x2, x1^2],
[x2, x1^2],
[x2, x1^2]]


The Singular documentation for ‘NF’ is given below.

5.1.127 reduce
--------------

*Syntax:*'
reduce (' poly_expression,' ideal_expression )'
reduce (' poly_expression,' ideal_expression,' int_expression
)'
reduce (' poly_expression,' poly_expression,' ideal_expression
)'
reduce (' vector_expression,' ideal_expression )'
reduce (' vector_expression,' ideal_expression,' int_expression
)'
reduce (' vector_expression,' module_expression )'
reduce (' vector_expression,' module_expression,'
int_expression )'
reduce (' vector_expression,' poly_expression,'
module_expression )'
reduce (' ideal_expression,' ideal_expression )'
reduce (' ideal_expression,' ideal_expression,' int_expression
)'
reduce (' ideal_expression,' matrix_expression,'
ideal_expression )'
reduce (' module_expression,' ideal_expression )'
reduce (' module_expression,' ideal_expression,' int_expression
)'
reduce (' module_expression,' module_expression )'
reduce (' module_expression,' module_expression,'
int_expression )'
reduce (' module_expression,' matrix_expression,'
module_expression )'
reduce (' poly/vector/ideal/module,' ideal/module,' int,'
intvec )'
reduce (' ideal,' matrix,' ideal,' int )'
reduce (' poly,' poly,' ideal,' int )'
reduce (' poly,' poly,' ideal,' int,' intvec )'

*Type:*'
the type of the first argument

*Purpose:*'
reduces a polynomial, vector, ideal  or module to its normal form
with respect to an ideal or module represented by a standard basis.
Returns 0 if and only if the polynomial (resp. vector, ideal,
module) is an element (resp. subideal, submodule) of the ideal
(resp. module).  The result may have no meaning if the second
argument is not a standard basis.
The third (optional) argument of type int modifies the behavior:
* 0 default

* 1 consider only the leading term and do no tail reduction.

* 2 tail reduction:n the local/mixed ordering case: reduce also

* 4 reduce without division, return possibly a non-zero
constant multiple of the remainder

If a second argument u' of type poly or matrix is given, the
first argument p' is replaced by p/u'.  This works only for zero
dimensional ideals (resp. modules) in the third argument and
gives, even in a local ring, a reduced normal form which is the
projection to the quotient by the ideal (resp. module).  One may
give a degree bound in the fourth argument with respect to a
weight vector in the fifth argument in order have a finite
computation.  If some of the weights are zero, the procedure may
not terminate!

*Note_*'
The commands reduce' and NF' are synonymous.

*Example:*'
ring r1 = 0,(z,y,x),ds;
poly s1=2x5y+7x2y4+3x2yz3;
poly s2=1x2y2z2+3z8;
poly s3=4xy5+2x2y2z3+11x10;
ideal i=s1,s2,s3;
ideal j=std(i);
reduce(3z3yx2+7y4x2+yx5+z12y2x2,j);
==> -yx5+2401/81y14x2+2744/81y11x5+392/27y8x8+224/81y5x11+16/81y2x14
reduce(3z3yx2+7y4x2+yx5+z12y2x2,j,1);
==> -yx5+z12y2x2
// 4 arguments:
ring rs=0,x,ds;
// normalform of 1/(1+x) w.r.t. (x3) up to degree 5
reduce(poly(1),1+x,ideal(x3),5);
==> // ** _ is no standard basis
==> 1-x+x2

See
* division::
* ideal::
* module::
* poly operations::
* std::
* vector::

sage.algebras.letterplace.free_algebra_letterplace.singular_system(ring=None, interruptible=True, attributes=None, *args)

This function is an automatically generated C wrapper around the Singular function ‘system’.

This wrapper takes care of converting Sage datatypes to Singular datatypes and vice versa. In addition to whatever parameters the underlying Singular function accepts when called, this function also accepts the following keyword parameters:

INPUT:

• args – a list of arguments
• ring – a multivariate polynomial ring
• interruptible – if True pressing Ctrl-C during the execution of this function will interrupt the computation (default: True)
• attributes – a dictionary of optional Singular attributes assigned to Singular objects (default: None)

If ring is not specified, it is guessed from the given arguments. If this is not possible, then a dummy ring, univariate polynomial ring over QQ, is used.

EXAMPLES:

sage: groebner = sage.libs.singular.function_factory.ff.groebner
sage: P.<x, y> = PolynomialRing(QQ)
sage: I = P.ideal(x^2-y, y+x)
sage: groebner(I)
[x + y, y^2 - y]
sage: triangL = sage.libs.singular.function_factory.ff.triang__lib.triangL
sage: P.<x1, x2> = PolynomialRing(QQ, order='lex')
sage: f1 = 1/2*((x1^2 + 2*x1 - 4)*x2^2 + 2*(x1^2 + x1)*x2 + x1^2)
sage: f2 = 1/2*((x1^2 + 2*x1 + 1)*x2^2 + 2*(x1^2 + x1)*x2 - 4*x1^2)
sage: I = Ideal(Ideal(f1,f2).groebner_basis()[::-1])
sage: triangL(I, attributes={I:{'isSB':1}})
[[x2^4 + 4*x2^3 - 6*x2^2 - 20*x2 + 5, 8*x1 - x2^3 + x2^2 + 13*x2 - 5],
[x2, x1^2],
[x2, x1^2],
[x2, x1^2]]


The Singular documentation for ‘system’ is given below.

5.1.151 system
--------------

*Syntax:*'
system (' string_expression )'
system (' string_expression,' expression )'

*Type:*'
depends on the desired function, may be none

*Purpose:*'
interface to internal data and the operating system. The
string_expression determines the command to execute. Some commands
require an additional argument (second form) where the type of the
argument depends on the command. See below for a list of all
possible commands.

*Note_*'
Not all functions work on every platform.

*Functions:*'

system("alarm",' int )'
abort the Singular process after computing for that many
seconds (system+user cpu time).

system("absFact",' poly )'
absolute factorization of the polynomial (from a polynomial
ring over a transzedental extension) Returns a list of the
ideal of the factors, intvec of multiplicities, ideal of
minimal polynomials and the bumber of factors.

system("blackbox")'
list all blackbox data types.

system("browsers");'
returns a string about available help browsers.  *Note The

system("bracket",' poly, poly )'
returns the Lie bracket [p,q].

system("btest",' poly, i2 )'
internal for shift algebra (with i2 variables): last block of
the poly

system("complexNearZero",' number_expression )'
checks for a small value for floating point numbers

system("contributors")'
returns names of people who contributed to the SINGULAR
kernel as string.

system("cpu")'
returns the number of cpus as int (for creating multiple
threads/processes).  (see system("--cpus")').

system("denom_list")'
returns the list of denominators (number) which occured in
the latest std computationi(s).  Is reset to the empty list
at ring changes or by this system call.

system("eigenvals",' matrix )'
returns the list of the eigenvalues of the matrix (as ideal,
intvec).  (see system("hessenberg")').

system("env",' ring )'
returns the enveloping algebra (i.e. R tensor R^opp) See
system("opp")'.

system("executable",' string )'
returns the path of the command given as argument or the
empty string (for: not found) See system("Singular")'.  See
system("getenv","PATH")'.

system("freegb",' ideal, i2, i3 )'
returns the standrda basis in the shift algebra i(with i3
variables) up to degree i2.  See system("opp")'.

system("getenv",' string_expression)'
returns the value of the shell environment variable given as
the second argument. The return type is string.

system("getPrecDigits")'
returns the precision for floating point numbers

system("gmsnf",' ideal, ideal, matrix,int, int )'
Gauss-Manin system: for gmspoly.lib, gmssing.lib

system("HC")'
returns the degree of the "highest corner" from the last std
computation (or 0).

system("hessenberg",' matrix )'
returns the Hessenberg matrix (via QR algorithm).

system("install",' s1, s2, p3, i4 )'
install a new method p3 for s2 for the newstruct type s1.  s2
must be a reserved operator with i4 operands (i4 may be
1,2,3; use 4 for more than 3 or a varying number of arguments)
See *Note Commands for user defined types::.

system("LLL",' B )'
B must be a matrix or an intmat.  Interface to NTLs LLL
(Exact Arithmetic Variant over ZZ).  Returns the same type as
the input.
B is an m x n matrix, viewed as m rows of n-vectors.  m may
be less than, equal to, or greater than n, and the rows need
not be linearly independent.  B is transformed into an
LLL-reduced basis.  The first m-rank(B) rows of B are zero.
More specifically, elementary row transformations are
performed on B so that the non-zero rows of new-B form an
LLL-reduced basis for the lattice spanned by the rows of
old-B.

system("nblocks")' or system("nblocks",' ring_name )'
returns the number of blocks of the given ring, or of the
current basering, if no second argument is given. The return
type is int.

system("nc_hilb",' ideal, int, [,...] )'
internal support for ncHilb.lib, return nothing

system("neworder",' ideal )'
string of the ring variables in an heurically good order for
char_series'

system("newstruct")'
list all newstruct data types.

system("opp",' ring )'
returns the opposite ring.

system("oppose",' ring R, poly p )'
returns the opposite polynomial of p from R.

system("pcvLAddL",' list, list )'
system("pcvPMulL",' poly, list )'
system("pcvMinDeg",' poly )'
system("pcvP2CV",' list, int, int )'
system("pcvCV2P",' list, int, int )'
system("pcvDim",' int, int )'
system("pcvBasis",' int, int )' internal for mondromy.lib

system("pid")'
returns the process number as int (for creating unique names).

system("random")' or system("random",' int )'
returns or sets the seed of the random generator.

system("reduce_bound",' poly, ideal, int )'
or system("reduce_bound",' ideal, ideal, int )'
or system("reduce_bound",' vector, module, int )'
or system("reduce_bound",' module, module, int )' returns
the normalform of the first argument wrt. the second up to
the given degree bound (wrt. total degree)

system("reserve",' int )'
reserve a port and listen with the given backlog.  (see
system("reservedLink")').

accept a connect at the reserved port and return a
(write-only) link to it.  (see system("reserve")').

system("semaphore",' string, int )'
operations for semaphores: string may be "init"', "exists"',
"acquire"', "try_acquire"', "release"', "get_value"', and
int is the number of the semaphore.  Returns -2 for wrong
command, -1 for error or the result of the command.

system("semic",' list, list )'
or system("semic",' list, list, int )' computes from list
of spectrum numbers and list of spectrum numbers the
semicontinuity index (qh, if 3rd argument is 1).

system("setenv",'string_expression, string_expression)'
sets the shell environment variable given as the second
argument to the value given as the third argument. Returns
the third argument. Might not be available on all platforms.

system("sh"', string_expression )'
shell escape, returns the return code of the shell as int.
The string is sent literally to the shell.

system("shrinktest",' poly, i2 )'
internal for shift algebra (with i2 variables): shrink the
poly

system("Singular")'
returns the absolute (path) name of the running SINGULAR as
string.

system("SingularLib")'
returns the colon seperated library search path name as
string.

system("spadd",' list, list )'
or system("spadd",' list, list, int )' computes from list
of spectrum numbers and list of spectrum numbers the sum of
the lists.

system("spectrum",' poly )'
or  system("spectrum",' poly, int )'

system("spmul",' list, int )'
or system("spmul",' list, list, int )' computes from list
of spectrum numbers the multiple of it.

system("std_syz",' module, int )'
compute a partial groebner base of a module, stopp after the
given column

system("stest",' poly, i2, i3, i4 )'
internal for shift algebra (with i4 variables): shift the
poly by i2, up to degree i3

system("tensorModuleMult",' int, module )'
internal for sheafcoh.lib (see id_TensorModuleMult)

system("twostd",' ideal )'
returns the two-sided standard basis of the two-sided ideal.

system("uname")'
returns a string identifying the architecture for which
SINGULAR was compiled.

system("version")'
returns the version number of SINGULAR as int.  (Version
a-b-c-d returns a*10000+b*1000+c*100+d)

system("with")'
without an argument: returns a string describing the current
version of SINGULAR, its build options, the used path names
and other configurations
with a string argument: test for that feature and return an
int.

system("--cpus")'
returns the number of available cpu cores as int (for using
multiple cores).  (see system("cpu")').

system("'-")'
prints the values of all options.

system("'-long_option_name")'
returns the value of the (command-line) option
long_option_name. The type of the returned value is either

system("'-long_option_name",' expression)'
sets the value of the (command-line) option long_option_name
to the value given by the expression. Type of the expression
must be string, or int. *Note Command line options::, for
seed of the random number generator, the used help browser,
the minimal display time, or the timer resolution.

*Example:*'
// a listing of the current directory:
system("sh","ls");
system("sh","sh");
string unique_name="/tmp/xx"+string(system("pid"));
unique_name;
==> /tmp/xx4711
system("uname")
==> ix86-Linux
system("getenv","PATH");
==> /bin:/usr/bin:/usr/local/bin
system("Singular");
==> /usr/local/bin/Singular
// report value of all options
system("--");
==> // --batch           0
==> // --execute
==> // --sdb             0
==> // --echo            1
==> // --profile         0
==> // --quiet           1
==> // --sort            0
==> // --random          12345678
==> // --no-tty          1
==> // --user-option
==> // --allow-net       0
==> // --browser
==> // --cntrlc
==> // --emacs           0
==> // --no-stdlib       0
==> // --no-rc           1
==> // --no-warn         0
==> // --no-out          0
==> // --no-shell        0
==> // --min-time        "0.5"
==> // --cpus            4
==> // --MPport
==> // --MPhost
==> // --ticks-per-sec   1
// set minimal display time to 0.02 seconds
system("--min-time", "0.02");
// set timer resolution to 0.01 seconds
system("--ticks-per-sec", 100);
// re-seed random number generator
system("--random", 12345678);
system("--allow-net", 1);
// and set help browser to firefox
system("--browser", "firefox");
==> // ** Could not get 'DataDir'.
==> // ** Either set environment variable 'SINGULAR_DATA_DIR' to 'DataDir',
==> // ** or make sure that 'DataDir' is at "/home/hannes/singular/doc/../Sin\
gular/../share/"
==> // ** Could not get 'IdxFile'.
==> // ** Either set environment variable 'SINGULAR_IDX_FILE' to 'IdxFile',
==> // ** Could not get 'DataDir'.
==> // ** Either set environment variable 'SINGULAR_DATA_DIR' to 'DataDir',
==> // ** or make sure that 'DataDir' is at "/home/hannes/singular/doc/../Sin\
gular/../share/"
==> // ** or make sure that 'IdxFile' is at "%D/singular/singular.idx"
==> // ** resource x not found
==> // ** Setting help browser to 'dummy'.
`