QAuxiliaryTools¶
qetatool.spad line 149 [edit on github]
Miscellaneous tools to deal with polynomials.
- clearDenominator: Polynomial Fraction Integer -> Polynomial Integer
- clearDenominator(p)multiplies the polynomial- pover rational numbers with the least common multiple of the denominator of all its coefficients and returns the result as a polynomial over the integers.
- fromEtaToQPochhammer: (NonNegativeInteger, Fraction Polynomial Integer, String, String) -> Record(frf: Fraction Polynomial Integer, texpo: Integer)
- fromEtaToQPochhammer(m,rf,e,u)returns [np/dp, ne-de] where [- np,ne]:=fromEtaToQPochhammer(- m,numer(- rf),- e,- u), [- dp,de]:=fromEtaToQPochhammer(- m,denom(- rf),- e,- u).
- fromEtaToQPochhammer: (NonNegativeInteger, Polynomial Integer, String, String) -> Record(fpol: Polynomial Integer, texpo: Integer)
- fromEtaToQPochhammer(m,p,e,u)is like the inverse of fromQPochhammerToEta. The variable- tcorresponds to exp(- pi*i*tau/12) and- q=t^24. fromEtaToPochhammer(- m,- p,- e,- u) returns [- r,- n] where- ris a polynomial in- qand the- uisuch that- pand r*t^n correspond to the same- q-series expansion.- ris 0 if- pcannot be expressed in such a way.
- fromQPochhammerToEta: (NonNegativeInteger, Fraction Polynomial Integer, String, String) -> Record(frf: Fraction Polynomial Integer, texpo: Integer)
- fromQPochhammerToEta(m,rf,u,e)returns [np/dp, ne-de] where [- np,ne]:=fromQPochhammerToEta(- m,numer(- rf),- u,- e), [- dp,de]:=fromQPochhammerToEta(- m,denom(- rf),- u,- e).
- fromQPochhammerToEta: (NonNegativeInteger, Polynomial Integer, String, String) -> Record(fpol: Polynomial Integer, texpo: Integer)
- fromQPochhammerToEta(m,p,u,e)expresses- p(given as a polynomial in variables- uiand- qwhere the- uicorrespond to the Euler functions of level- mhttps://en.wikipedia.org/wiki/Euler_function) into an expression in variables- ei(corresponding to eta(i*tau)). The result [- r,- n] is expressed as a polynomial- rin the- eiand an exponent- nof- t(corresponding to exp(- pi*i*tau/12)) such that- pand r*t^n correspond to the same- q-series expansion (with- q=t^24).- ris 0 if- pcannot be expressed in such a way. This function mainly helps to translate the representation of Somos at url{https://web.archive.org/web/20190709153133/http://eta.math.georgetown.- edu/eta07.- gp} in terms of Euler functions to our representation in terms of eta functions. The variable- qcorresponds to exp(2*pi*i*tau).
- index: (String, String) -> List Integer
- index(s,i)returns the list integers encoded in the string- s(and separated by one underscore) when the initial string- iis removed. The function returns the empty list if the string does not start with the initial or has not the right format. It can be considered as the inverse of indexedSymbol.
- indexedSymbol: (String, List Integer) -> Symbol
- indexedSymbol(s, l)returns a symbol with a name that starts with the string- sand is directly followed by the numbers of- lseparated by underscores (“__”).
- indexedSymbols: (String, List Integer) -> List Symbol
- indexedSymbols(s, l)returns indexedSymbols(- s, [[- n] for- nin- l]).
- indexedSymbols: (String, List List Integer) -> List Symbol
- indexedSymbols(s, ll)returns [indexedSymbol(- s,- l) for- lin- ll].
- indexedSymbols: (String, NonNegativeInteger) -> List Symbol
- indexedSymbols(s, n)returns indexedSymbols(- s, [- ifor- iin 1..- n])
- integerPrimitivePart: Polynomial Fraction Integer -> Polynomial Integer
- integerPrimitivePart(p)returns primitivePart(clearDenominator(- p)).