Invariants and transformations

Invariant class that associates a value with every dimension vector

An invariant can be constructed from a TMPolynomial, a dictionary, a function or another invariant. Every invariant can be transformed to a TMPolynomial or a dictionary. We cache the values of an invariant so that they don’t have to be computed repeatedly.

EXAMPLES:

sage: from msinvar import *
sage: R=TMPoly(prec=[3]); R
Multivariate Polynomial Ring in x over Rational Field truncated at degree [3]
sage: x=R.gen(); x.Exp()
1 + x + x^2 + x^3
sage: I=Invariant(x.Exp()); I
Invariant
sage: I.dict()
{(0,): 1, (1,): 1, (2,): 1, (3,): 1}
sage: I.Log().dict()
{(1,): 1}
sage: I.Log().poly()
x

class msinvar.invariants.Invariant(f, R=None)

Bases: object

Invariant class that associates an invariants I(d) with every dimension vector d.

An argument can be a TMPolynomial, a dictionary, a function or another invariant. Objects have methods that return associated TMPolynomials and dictionaries. If possible, we save the ring (TMPoly) in order to transform our invariant to a polynomial.

• f – function, TMPolynomial, dictionary, Invariant.

• R – a ring TMPolynomialRing.

value()

get_ring(I=None)

The ring (TMPolynomialRing) saved by the invariant.

get_prec(d=None)

Precision vector of the ring.

dict(prec=None, stab=None, slope=0)

Convert invariant to a dictionary, with keys bounded by self.prec().

poly(R=None, stab=None, slope=None)

Convert invariant to a TMPolynomial.

to_series(R=None, stab=None, slope=None)

Convert invariant to a TMPolynomial.

series(R=None, stab=None, slope=None)

Convert invariant to a TMPolynomial.

term_twist(f)

Twist every term using the function f.

restrict(z, slope)

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

subs(**kw)

Apply substitution to every term.

root_vars(k=2)

Substitute y^k by y for every variable y.

simp(d=None)

Simplify every term.

plog(algo='fast')

Take log along every ray.

pexp(algo='fast')

Take exp along every ray.

exp()

The usual exponential. This method can be quite slow.

log()

The usual logarithm. This method can be quite slow.

Psi()

Plethystic map Psi.

IPsi()

Plethystic map Psi inverse.

pLog()

Take plethystic Log along every ray.

pExp()

Take plethystic Exp along every ray.

Exp()

Take plethystic Exp.

Log()

Take plethystic Log.

msinvar.invariants.log_map(I, d=None)

Usual log.

msinvar.invariants.exp_map(I, d=None)

Usual exp.

msinvar.invariants.plog_map(I, d=None)

log along every ray.

msinvar.invariants.pexp_map(I, d=None)

exp along every ray.

msinvar.invariants.Psi_map(I, d=None)

Plethystic Psi.

msinvar.invariants.IPsi_map(I, d=None)

Plethsytic Psi inverse.

msinvar.invariants.rat2int_num(I, d=None)

msinvar.invariants.int2rat_num(I, d=None)

msinvar.invariants.recursive_inversion(T)

Invert the transformation (map on invariants) T, assuming that T(I)(d) is a sum of I(d) and some expression that depends just on I(e) with e<d.

Note that exp and log are not of this form for d=0.

class msinvar.invariants.Transform(F, twist=None)

Bases: object

A class to transform invariants. Based on arXiv:2101.07636.

It is encoded by a map from the set of lists of vectors to rational numbers.

We define actions of transforms on 1-collections (Invariant class), plethysm between transforms, inverse transfroms.

INPUT:

• F – transformation map from lists of vectors to Q (or base ring).

• twist – product twist, map from sequences of vectors to the base ring.

value()

action(I, d=None)

Action of the transformation on an invariant I.

plethysm(T, l=None)

Plethysm of self and T.

inverse()

dict(d)

Dictionary of values for all lists of vectors with the sum <=d.

msinvar.invariants.proportional(a, b=None)

Check if vectors a,b are proportional.

msinvar.invariants.exp_transform(z=None)

msinvar.invariants.log_transform(z=None)

msinvar.invariants.reineke_sign(l, z, tot=None)

Used in reineke_transform().

msinvar.invariants.reineke_transform(z)

Based on arXiv:math/0204059 and arXiv:2101.07636.

msinvar.invariants.joyce_sign(l, z, z1, tot=None)

Used in joyce_transform().

msinvar.invariants.joyce_transform(z, z1)

Based on arXiv:math/0410268 and arXiv:2101.07636.

msinvar.invariants.HN_transform(z)

See arXiv:2101.07636.

msinvar.invariants.indivisible(d)

Return True if d is an indivisible vector.

msinvar.invariants.plog_fast(I)

Return the logarithm taken separately along each ray.

We construct a dictionary of such logarithms with keys parametrized by indivisible vectors. The logarithm is taken in the ring of truncated polynomials. Therefore I.R and its precision vector are required.

msinvar.invariants.pexp_fast(I)

See plog_fast().