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13. Polynomials (Standard Mathematical Functions)


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13.0.1 poly_degree

poly_degree(F)

: It returns the degree of F with respect to the given weight vector.

poly_degree(F | weight=key0,v=key1)

: This function allows optional variables weight, v

Description:

The weight is given by the optional variable weight w. It returns

Example:

 
 poly_degree(x^2+y^2-4 |weight=[100,1],v=[x,y]);

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13.0.2 poly_elimination_ideal

poly_elimination_ideal(I,VV)

: It computes the ideal intersection of I and the monomial ideal generated by VV.

poly_elimination_ideal(I,VV | grobner_basis=key0,v=key1)

: This function allows optional variables grobner_basis, v

Description:

If grobner_basis is "yes", I is assumed to be a Grobner basis. The optional variable v is a list of variables which defines the ring of polynomials.

Example 0:

 
 poly_elimination_ideal([x^2+y^2-4,x*y-1],[x]);

Example 1:

 
 A = poly_grobner_basis([x^2+y^2-4,x*y-1]|order=2,v=[y,x]);
          poly_elimination_ideal(A,[x]|grobner_basis="yes");

References:

gr , hgr , gr_mod , dp_*


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13.0.3 poly_expand

poly_expand(F)

: This is an alias of poly_sort.

References:

poly_sort


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13.0.4 poly_factor

poly_factor(F)

: It factorizes the polynomial F.

Example:

 
 poly_factor(x^10-y^10);

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13.0.5 poly_gcd

poly_gcd(F,G)

: It computes the polynomial GCD of F and G.

Example:

 
 poly_gcd(x^10-y^10,x^25-y^25);

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13.0.6 poly_grobner_basis

poly_grobner_basis(I)

: It returns the Grobner basis of I.

poly_grobner_basis(I | order=key0,v=key1)

: This function allows optional variables order, v

Description:

The optional variable v is a list of variables which defines the ring of polynomials.

Example:

 
 A = poly_grobner_basis([x^2+y^2-4,x*y-1]|order=2,v=[y,x]);

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13.0.7 poly_hilbert_polynomial

poly_hilbert_polynomial(I)

: It returns the Hilbert polynomial of the ideal I.

poly_hilbert_polynomial(I | s=key0,v=key1)

: This function allows optional variables s, v

Description:

The optional variable v is a list of variables.

Example:

 
 poly_hilbert_polynomial([x1*y1,x1*y2,x2*y1,x2*y2]|s=k,v=[x1,x2,y1,y2]);

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13.0.8 poly_initial

poly_initial(I)

: It returns the initial ideal of I with respect to the given order.

poly_initial(I | order=key0,v=key1)

: This function allows optional variables order, v

Description:

The optional variable v is a list of variables. This function computes

Example:

 
 poly_initial([x^2+y^2-4,x*y-1]|order=0,v=[x,y]);

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13.0.9 poly_initial_coefficients

poly_initial_coefficients(I)

: It computes the coefficients of the initial ideal of I with respect to the given order.

poly_initial_coefficients(I | order=key0,v=key1)

: This function allows optional variables order, v

Description:

The optional variable v is a list of variables. The order is specified by the optional variable order

Example:

 
 poly_initial_coefficients([x^2+y^2-4,x*y-1]|order=0,v=[x,y]);

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13.0.10 poly_initial_term

poly_initial_term(F)

: It returns the initial term of a polynomial F with respect to the given weight vector.

poly_initial_term(F | weight=key0,order=key1,v=key2)

: This function allows optional variables weight, order, v

Description:

The weight is given by the optional variable weight w. It returns

Example:

 
 poly_initial_term( x^2+y^2-4 |weight=[100,1],v=[x,y]);

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13.0.11 poly_solve_linear

poly_solve_linear(Eqs,V)

: It solves the system of linear equations Eqs with respect to the set of variables V.

Example:

 
 poly_solve_linear([2*x+3*y-z-2, x+y+z-1], [x,y,z]);

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13.0.12 poly_sort

poly_sort(F)

: It expands F with a given variables v=V and a given weight w=W. It returns a quote object. If trucate option is set, the expansion is truncated at the given degree.

poly_sort(F | v=key0,w=key1,truncate=key2)

: This function allows optional variables v, w, truncate

Example:

 
 poly_sort((x-y-a)^3 | v=[x,y], w=[-1,-1])  
    returns a series expansion in terms of x and y.

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This document was generated by Nobuki Takayama on January, 28 2008 using texi2html 1.76.