Free associative unital algebras, implemented via Singular’s letterplace rings¶
AUTHOR:
Simon King (2011-03-21): trac ticket #7797
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 cannot 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.
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class
sage.algebras.letterplace.free_algebra_letterplace.
FreeAlgebra_letterplace
¶ Bases:
sage.rings.ring.Algebra
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
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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
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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
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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
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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
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generator_degrees
()¶
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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
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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
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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 cannot be constructed in this implementation.
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ngens
()¶ Return the number of generators.
EXAMPLES:
sage: F.<a,b,c> = FreeAlgebra(QQ, implementation='letterplace') sage: F.ngens() 3
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set_degbound
(d)¶ Increase the degree bound that is currently in place.
NOTE:
The degree bound cannot 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
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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
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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 argumentsring
– a multivariate polynomial ringinterruptible
– ifTrue
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 overQQ
, 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 with bad ecart * 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 * Menu: See * division:: * ideal:: * module:: * poly operations:: * std:: * vector::
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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 argumentsring
– a multivariate polynomial ringinterruptible
– ifTrue
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 overQQ
, 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 online help system::. `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")'). `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 string or int. *Note Command line options::, for more info. `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 more info. Among others, this can be used for setting the 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"); // execute a shell, return to SINGULAR with exit: 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 ==> // --threads 4 ==> // --MPport ==> // --MPhost ==> // --link ==> // --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); // allow your web browser to access HTML pages from the net 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'.