Truncated multivariate polynomials

Truncated Multivariate polynomials

EXAMPLES:

sage: from msinvar import TMPoly
sage: R=TMPoly(QQ,2,'x',prec=(2,2))
sage: R.inject_variables(verbose=False)
sage: (x0+x1).Exp()
1 + x0 + x1 + x0^2 + x0*x1 + x1^2 + x0^2*x1 + x0*x1^2 + x0^2*x1^2

class msinvar.tm_polynomials.TMPolynomial(parent, x)

Bases: sage.rings.polynomial.multi_polynomial_element.MPolynomial_polydict

The element class for truncated multivariate polynomials. The parent class is TMPolynomialRing.

Implementation details:

1. The parent should have a method le_prec() to test if a given degree should be present.

2. Truncation is performed in the __init__() method.

3. Arithmetic operations are inherited from MPolynomial_polydict.

4. We implement Adams operation adams, exp, log and plethystic Exp, Log.

5. Function for the twisted multilpication is given by self.parent().prod_twist

le_prec(v)

prec_num()

exp()

log()

invert()

For simplicity we invert only poly with constant coefficient 1

Psi()

IPsi()

Exp()

Log()

term_twist(f)

Multiply every term c*x^v by f(v)

coeff(e)

Return the coefficient of degree e monomial as an element of the base_ring.

Note that coefficient returns an element of the polynomial ring.

subs_base(**kw)

subs(**kw)

Fixes some given variables in a given multivariate polynomial and returns the changed multivariate polynomials. The polynomial itself is not affected. The variable,value pairs for fixing are to be provided as a dictionary of the form {variable:value}.

This is a special case of evaluating the polynomial with some of the variables constants and the others the original variables.

INPUT:

• fixed - (optional) dictionary of inputs

• **kw - named parameters

OUTPUT: new MPolynomial

EXAMPLES:

sage: R.<x,y> = QQbar[]
sage: f = x^2 + y + x^2*y^2 + 5
sage: f((5,y))
25*y^2 + y + 30
sage: f.subs({x:5})
25*y^2 + y + 30

restrict(z, slope)

Restrict polynomial to dimension vectors d such that z(d)=slope, where z is a Stability (or the corresponding vector).

invar()

pLog()

Plethystic logarithm along every ray.

pExp()

Plethystic logarithm along every ray.

class msinvar.tm_polynomials.TMPolynomialRing(base_ring=Rational Field, n=1, names='x', order='negdegrevlex', prec=None, prod_twist=None)

Bases: sage.rings.polynomial.multi_polynomial_ring.MPolynomialRing_polydict

Truncated Multivariate polynomial ring.

Multivariate polynomials up to degree prec, where prec is a degree vector or an integer (total degree). Monomials of degree prec are included. We implement Adams operations, plethystic Exp and Log (we introduce the category of lambda rings msinvar.lambda_rings.LambdaRings and embed our ring into it).

This is the parent class for truncated multivariate polynomials. It has an alias TMPoly. The element class is TMPolynomial.

PARAMETERS:

1. base_ring – the base of our polynomial algebra.

2. n – the number of vairables.

3. names – names of variables; can be just ‘x’ or ‘x,y,..’.

4. prec - precision vector (or integer, or None) for truncation

5. order – the order of variables.

6. prod_twist - function that maps exponents (d,e) to the base_ring. The new product is x^d*x^e=prod_twist(d,e)*x^(d+e)

EXAMPLES:

sage: from msinvar import TMPoly
sage: R=TMPoly(QQ,2,prec=(2,2)); R
Multivariate Polynomial Ring in x0, x1 over Rational Field truncated at degree (2, 2)
sage: x=R.gens(); (x[0]+x[1])**3
3*x0^2*x1 + 3*x0*x1^2

sage: QR=Frac(PolynomialRing(QQ,'y,t'))
sage: S=TMPoly(QR,2,'x',prec=(2,2))
sage: y,t=QR.gens(); x=S.gens()
sage: (y*x[0]+x[1]).adams(2)
y^2*x0^2 + x1^2
sage: (y*x[0]).Exp()
1 + y*x0 + y^2*x0^2

sage: R=TMPoly(QQ,1,prec=2, prod_twist=lambda a,b:2)
sage: x=R.gen(); x*x
2*x^2

Element

alias of msinvar.tm_polynomials.TMPolynomial

adams(a, n)

zero_tuple()

prec(d=None)

prec_num()

le_prec(v)

msinvar.tm_polynomials.TMPoly

alias of msinvar.tm_polynomials.TMPolynomialRing

msinvar.tm_polynomials.Exp(f)

msinvar.tm_polynomials.Log(f)

msinvar.tm_polynomials.Psi(f)

msinvar.tm_polynomials.IPsi(f)