| Extra Algebra | |
| Functions | |
| PolynomialSaturation | Saturates an ideal by a polynomial. |
| SizeReduction | Reduces the number of polynomials. |
| IsHullPrime | Test whether the maximal dimension component is prime. |
| IsHullRadical | Test whether the maximal dimension component is radical. |
| IsProjectionProper | Sufficient condition for proper projection. |
| PropernessDefect | Computes the non properness set of a projection. |
| SortPolynomials | Sort a list of polynomial and trigonometric expressions |
| FactorSystem | Factorize partially a system |
| Hypersurfaces | Factorizes components of codimension 1. |
| RandomSection | Cuts a system with a random hypersurface |
| ToLatex | Print the latex commands generating an expression on multiple lines. |
PolynomialSaturation := proc ( sys ::list(algebraic), p ::algebraic, vars ::list(name) := NULL, char ::integer := 0 )
Saturates an ideal by a polynomial.
| sys | list of polynomials: the generators of the ideal |
| p | polynomial with rational coefficients: the saturating polynomial |
| vars (optional) | list of names: the variables in sys and p |
| char (optional) | integer: characteristic of the base field default value: 0 |
| verbose (optional) | ”verb”=true or “verb”=false: verbose support default value: “verb”=false |
A list of polynomials: the generators of sys saturated by p.
> PolynomialSaturation ( [x^2*y,z], x ); < [y,z]
SizeReduction := proc ( sys ::list(algebraic), vars ::list(name) := NULL )
Reduces the number of polynomials.
| sys | list of polynomials: the system to reduce |
| vars (optional) | list of names: the variables in sys |
A list of polynomials: a subset of sys with the same zeroes
> SizeReduction ([x,x]); < [x]
IsHullPrime := proc ( sys ::list(algebraic), vars ::list(name) := [ op(indets(sys,name)) ] )
Test whether the maximal dimension component is prime.
| sys | list of polynomials with rational coefficients: the system |
| vars (optional) | list of names: variables of the system |
A boolean: true means that the maximal component of the ideal generated by sys is prime, while false means that it may be not prime.
> IsHullPrime ([x*z,y*z]); < true
IsHullRadical := proc ( sys ::list(algebraic), pars ::list(name) := NULL, vars ::list(name) := [ op(indets(sys,name)) ] )
Test whether the maximal dimension component is radical.
| sys | list of polynomials with rational coefficients: the system |
| pars (optional) | list of names: maximal independant set of variables |
| vars (optional) | list of names: variables of the system |
A boolean: true if the maximal component of the ideal generated by sys is radical, false otherwise.
> IsHullPrime ([x*z,y*z]); < true
IsProjectionProper := proc ( sys ::list(polynom), pars ::list(name), vars ::list(name) := [op(indets(sys, name) minus {op(pars)})] )
Sufficient condition for proper projection.
| sys | list of polynomials: the equations |
| pars | list of names: the image of the projection; |
| vars (optional) | list of names: the other variables default value: the variables in sys not in pars |
The projection is proper if the function returns true, unknown otherwise.
PropernessDefect := proc ( sys ::list(algebraic), pars ::list(name), vars ::list(name) := [op(indets(sys, name) minus {op(pars)})], char ::integer := 0 )
Computes the non properness set of a projection.
| sys | list of polynomials: the equations |
| pars | list of names: the image of the projection; |
| vars (optional) | list of names: the other variables default value: the variables in sys not in pars |
| char (optional) | integer: the characteristic of the coefficient field |
A list of list of polynomials: each list is a list of polynomials in pars, the zeroes of which is a component of the properness defect set.
SortPolynomials := proc ( sys ::list(algebraic), ordering ::{ShortMonomialOrder, list(name)} := NULL )
Sort a list of polynomial and trigonometric expressions
| sys | list of polynomial or trigonometric expressions |
| ordering (optional) | a monomial ordering given as a ShortMonomialOrder or a list of name (in this case, the lexicographic ordering is used) |
The sorted list of polynomial and trigonometric expressions.
FactorSystem := proc ( sys ::{list(algebraic),list(list(algebraic))}, { ordering ::list(name) := NULL } )
Factorize partially a system
| sys | list or list of list of algebraic expressions; the system to factorize |
A list of list of algebraic expressions: each list is a factor of the input system
Hypersurfaces := proc ( sys ::list(algebraic), { squarefree ::truefalse := true } )
Factorizes components of codimension 1.
| sys | a list of polynomials: the sytem to decompose |
| squarefree = b (optional) | b is a boolean: if b is false, then the multiplicities are saved; default value: true |
A list of list of polynomials: the different components of the system. The multiplicities of the different components are preserved.
Hypersurfaces ([x^2*y, x*z]);
< [ [x], [ x^2*y, z] ]
Print the latex commands generating an expression on multiple lines.
| a | Mathematical expression to print. |
| k (optional) | integer, the width of the lines; default value: 0. |
| pos (optional) | either ‘b’ or ‘t’: when ‘b’ (resp. ‘t’), the surrounding text is alined with the bottom of the expression; default value: ‘t’. |
A latex version of the formula exp spreading on multiple lines of width roughly equal to k, with the position pos.
Saturates an ideal by a polynomial.
PolynomialSaturation := proc ( sys ::list(algebraic), p ::algebraic, vars ::list(name) := NULL, char ::integer := 0 )
Reduces the number of polynomials.
SizeReduction := proc ( sys ::list(algebraic), vars ::list(name) := NULL )
Test whether the maximal dimension component is prime.
IsHullPrime := proc ( sys ::list(algebraic), vars ::list(name) := [ op(indets(sys,name)) ] )
Test whether the maximal dimension component is radical.
IsHullRadical := proc ( sys ::list(algebraic), pars ::list(name) := NULL, vars ::list(name) := [ op(indets(sys,name)) ] )
Sufficient condition for proper projection.
IsProjectionProper := proc ( sys ::list(polynom), pars ::list(name), vars ::list(name) := [op(indets(sys, name) minus {op(pars)})] )
Computes the non properness set of a projection.
PropernessDefect := proc ( sys ::list(algebraic), pars ::list(name), vars ::list(name) := [op(indets(sys, name) minus {op(pars)})], char ::integer := 0 )
Sort a list of polynomial and trigonometric expressions
SortPolynomials := proc ( sys ::list(algebraic), ordering ::{ShortMonomialOrder, list(name)} := NULL )
Factorize partially a system
FactorSystem := proc ( sys ::{list(algebraic),list(list(algebraic))}, { ordering ::list(name) := NULL } )
Factorizes components of codimension 1.
Hypersurfaces := proc ( sys ::list(algebraic), { squarefree ::truefalse := true } )
Cuts a system with a random hypersurface
RandomSection := proc ( sys ::list(algebraic), vars ::list(name) := [op(indets(sys,name))] )