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Translation from Somos notation to eta-function notation and back

Related to http://www.risc.jku.at/people/hemmecke/eta/ and http://axiom-wiki.newsynthesis.org/EtaRelations8 .

We demonstrate below how :

    t8_12_24 = +1*u1^8*u4^4 +8*q*u1^4*u2^2*u4^2*u8^4 -1*u2^12

can be translated into :

    8*E1^4*E2^2*E4^2*E8^4+E1^8*E4^4-E2^12

spad
-------------------------------------------------------------------
---
--- FriCAS QEta
--- Copyright (C) 2015-2017  Ralf Hemmecke <ralf@hemmecke.org>
---
-------------------------------------------------------------------
-- This program is free software: you can redistribute it and/or modify
-- it under the terms of the GNU General Public License as published by
-- the Free Software Foundation, either version 3 of the License, or
-- (at your option) any later version.
--
-- This program is distributed in the hope that it will be useful,
-- but WITHOUT ANY WARRANTY; without even the implied warranty of
-- MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
-- GNU General Public License for more details.
--
-- You should have received a copy of the GNU General Public License
-- along with this program.  If not, see <http://www.gnu.org/licenses/>.
-------------------------------------------------------------------
OF==>OutputForm dbgPrint(x,y) ==> print([x::OF, y::OF]$List(OF)::OF) rep x ==> (x@%) pretend Rep per x ==> (x@Rep) pretend % P ==> PositiveInteger N ==> NonNegativeInteger Z ==> Integer Q ==> Fraction Z UP ==> SparseUnivariatePolynomial C SQ ==> SquareMatrix(n, Q) DQ ==> DirectProduct(n, Q) VZ ==> Vector Z VQ ==> Vector Q LSym ==> List Symbol
)abbrev package POLYEVAL PolynomialEvaluation PolynomialEvaluation(_ C: Ring, _ S: with (_ _+: (%, %) -> %; _ _*: (%, %) -> %; _ _^: (%, NonNegativeInteger) -> %)_ ): with eval: (Polynomial C, C -> S, LSym, List S) -> S == add eval(p: Polynomial C, embed: C -> S, vars: LSym, vals: List S): S == E ==> IndexedExponents Symbol PE ==> PolynomialCategoryLifting(E, Symbol, C, Polynomial C, S) map((s:Symbol):S +-> vals.position(s, vars), embed, p)$PE
)abbrev package STOETA SomosToEta SomosToEta: Exports == Implementation where Pol ==> Polynomial Z LPol ==> List Pol Exports ==> with toEta: (N, Pol) -> Pol ++ toEta(p) expresses p (given as a polynomial in variables ui and q ++ where the ui correspond to the Euler function ++ https://en.wikipedia.org/wiki/Euler_function) into an expression ++ in variables Ei (corresponding to eta(i*tau)). fromEta: (N, Pol) -> Pol ++ fromEta(p) expresses a polynomial p in variables Ei ++ (corresponding to eta(i*tau)) in terms of variables q and ui ++ where the ui correspond to the Euler function ++ https://en.wikipedia.org/wiki/Euler_function). Implementation ==> add indexedSymbols(s: String, n: N): List Symbol == [concat(s, convert(i)@String)::Symbol for i in 1..n] indexedSymbols(s: String, l: List Z): List Symbol == [concat(s, convert(i)@String)::Symbol for i in l] -- toEta(level, p) works by substituting E_d*t^d for u_d and -- t^(-24) for q. This gives a polynomial in t and the E -- variables. If everything is OK, the result should be a -- polynomial in the E variables times a power of t. We only -- return the factor that does not involve t. toEta(level: N, p: Pol): Pol == import from QAuxiliaryTools FPol ==> Fraction Pol divs: List Z := divisors(level)$IntegerNumberTheoryFunctions usyms: LSym := indexedSymbols("u", divs) esyms: LSym := indexedSymbols("E", divs) fc(c: Z): FPol == c::FPol syms: LSym := cons("q"::Symbol, usyms) evals: List FPol := [e::Pol::FPol for e in esyms] symt: Symbol := "t"::Symbol t: Pol := symt::Pol tt := t::FPol -- 1/q = t^24 vals: List FPol := [e*tt^d for e in evals for d in divs] vals := cons(inv(tt^24), vals) z: FPol := eval(p, fc, syms, vals)$PolynomialEvaluation(Z, FPol) if not one? denom z then error "toEta: denominator is not 1" x: Pol := numer z z: Union(Pol, "failed") := x exquo t k: N := 0 while z case Pol repeat k := k + 1 x := z :: Pol z := x exquo t dbgPrint("toEta: power of t", k) if member?(symt, variables x) then dbgPrint("debug message: toEta$CheckSomos contains t", x) return x
fromEta(level: N, p: Pol): Pol == import from QAuxiliaryTools divs: List Z := divisors(level)$IntegerNumberTheoryFunctions usyms: LSym := indexedSymbols("u", divs) esyms: LSym := indexedSymbols("E", divs) fc(c: Z): Pol == c::Pol uvals: LPol := [u::Pol for u in usyms] symt: Symbol := "t"::Symbol t: Pol := symt::Pol vals: LPol := [u*t^(qcoerce(d)@N) for u in uvals for d in divs] x: Pol := eval(p, fc, esyms, vals)$PolynomialEvaluation(Z, Pol) return x
spad
   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/1356809960828652341-25px001.spad
      using old system compiler.
   POLYEVAL abbreviates package PolynomialEvaluation 
------------------------------------------------------------------------
   initializing NRLIB POLYEVAL for PolynomialEvaluation 
   compiling into NRLIB POLYEVAL 
   compiling exported eval : (Polynomial C,C -> S,List Symbol,List S) -> S
   processing macro definition E ==> IndexedExponents Symbol 
   processing macro definition PE ==> PolynomialCategoryLifting(IndexedExponents Symbol,Symbol,C,Polynomial C,S) 
Time: 0.03 SEC.
(time taken in buildFunctor: 0)
;;; *** |PolynomialEvaluation| REDEFINED
;;; *** |PolynomialEvaluation| REDEFINED Time: 0 SEC.
Warnings: [1] eval: not known that (SetCategory) is of mode (CATEGORY domain (SIGNATURE + ($ $ $)) (SIGNATURE * ($ $ $)) (SIGNATURE ^ ($ $ (NonNegativeInteger))))
Cumulative Statistics for Constructor PolynomialEvaluation Time: 0.03 seconds
finalizing NRLIB POLYEVAL Processing PolynomialEvaluation for Browser database: --->-->PolynomialEvaluation(constructor): Not documented!!!! --->-->PolynomialEvaluation((+ (% % %))): Not documented!!!! --->-->PolynomialEvaluation((* (% % %))): Not documented!!!! --->-->PolynomialEvaluation((^ (% % (NonNegativeInteger)))): Not documented!!!! --->-->PolynomialEvaluation((eval (S (Polynomial C) (Mapping S C) (List (Symbol)) (List S)))): Not documented!!!! --->-->PolynomialEvaluation(): Missing Description ; compiling file "/var/aw/var/LatexWiki/POLYEVAL.NRLIB/POLYEVAL.lsp" (written 24 FEB 2018 04:49:36 AM):
; /var/aw/var/LatexWiki/POLYEVAL.NRLIB/POLYEVAL.fasl written ; compilation finished in 0:00:00.038 ------------------------------------------------------------------------ PolynomialEvaluation is now explicitly exposed in frame initial PolynomialEvaluation will be automatically loaded when needed from /var/aw/var/LatexWiki/POLYEVAL.NRLIB/POLYEVAL
STOETA abbreviates package SomosToEta ------------------------------------------------------------------------ initializing NRLIB STOETA for SomosToEta compiling into NRLIB STOETA compiling local indexedSymbols : (String,NonNegativeInteger) -> List Symbol Time: 0.02 SEC.
compiling local indexedSymbols : (String,List Integer) -> List Symbol Time: 0 SEC.
compiling exported toEta : (NonNegativeInteger,Polynomial Integer) -> Polynomial Integer processing macro definition FPol ==> Fraction Polynomial Integer Local variable z type redefined: (Union (Polynomial (Integer)) failed) to (Fraction (Polynomial (Integer))) Time: 0.08 SEC.
compiling exported fromEta : (NonNegativeInteger,Polynomial Integer) -> Polynomial Integer Time: 0.01 SEC.
(time taken in buildFunctor: 0)
;;; *** |SomosToEta| REDEFINED
;;; *** |SomosToEta| REDEFINED Time: 0 SEC.
Cumulative Statistics for Constructor SomosToEta Time: 0.11 seconds
finalizing NRLIB STOETA Processing SomosToEta for Browser database: --->-->SomosToEta(constructor): Not documented!!!! --------(toEta ((Polynomial (Integer)) (NonNegativeInteger) (Polynomial (Integer))))--------- --------(fromEta ((Polynomial (Integer)) (NonNegativeInteger) (Polynomial (Integer))))--------- --->-->SomosToEta((fromEta ((Polynomial (Integer)) (NonNegativeInteger) (Polynomial (Integer))))): Missing left pren "\\spad{fromEta(p)} expresses a polynomial \\spad{p} in variables \\spad{Ei} (corresponding to eta(i*tau)) in terms of variables \\spad{q} and \\spad{ui} where the \\spad{ui} correspond to the Euler function https://en.wikipedia.org/wiki/Euler_function)." --->-->SomosToEta(): Missing Description ; compiling file "/var/aw/var/LatexWiki/STOETA.NRLIB/STOETA.lsp" (written 24 FEB 2018 04:49:36 AM):
; /var/aw/var/LatexWiki/STOETA.NRLIB/STOETA.fasl written ; compilation finished in 0:00:00.058 ------------------------------------------------------------------------ SomosToEta is now explicitly exposed in frame initial SomosToEta will be automatically loaded when needed from /var/aw/var/LatexWiki/STOETA.NRLIB/STOETA

fricas
t8_12_24 := u1^8*u4^4 +8*q*u1^4*u2^2*u4^2*u8^4 -1*u2^12

\label{eq1}{8 \  q \ {{u 1}^{4}}\ {{u 2}^{2}}\ {{u 4}^{2}}\ {{u 8}^{4}}}+{{{u 1}^{8}}\ {{u 4}^{4}}}-{{u 2}^{12}}(1)
Type: Polynomial(Integer)
fricas
T8_12_24 := toEta(8, t8_12_24)
["toEta: power of t", 24]

\label{eq2}{8 \ {{E 1}^{4}}\ {{E 2}^{2}}\ {{E 4}^{2}}\ {{E 8}^{4}}}+{{{E 1}^{8}}\ {{E 4}^{4}}}-{{E 2}^{12}}(2)
Type: Polynomial(Integer)
fricas
A8_12_24 := fromEta(8, T8_12_24)

\label{eq3}{8 \ {{t}^{48}}\ {{u 1}^{4}}\ {{u 2}^{2}}\ {{u 4}^{2}}\ {{u 8}^{4}}}+{{{t}^{24}}\ {{u 1}^{8}}\ {{u 4}^{4}}}-{{{t}^{24}}\ {{u 2}^{12}}}(3)
Type: Polynomial(Integer)

The variable t is such that t^{24}=q.

fricas
B8_12_24 := eval(A8_12_24, t=q^(1/24))::Polynomial(Integer)

\label{eq4}{8 \ {{q}^{2}}\ {{u 1}^{4}}\ {{u 2}^{2}}\ {{u 4}^{2}}\ {{u 8}^{4}}}+{q \ {{u 1}^{8}}\ {{u 4}^{4}}}-{q \ {{u 2}^{12}}}(4)
Type: Polynomial(Integer)
fricas
C8_12_24 := factorList factor B8_12_24

\label{eq5}\begin{array}{@{}l}
\displaystyle
\left[{\left[{flg = \verb#"prime"#}, \:{fctr = q}, \:{xpnt = 1}\right]}, \: \right.
\
\
\displaystyle
\left.{
\begin{array}{@{}l}
\displaystyle
\left[{flg = \verb#"prime"#}, \: \right.
\
\
\displaystyle
\left.{fctr ={{8 \  q \ {{u 1}^{4}}\ {{u 2}^{2}}\ {{u 4}^{2}}\ {{u 8}^{4}}}+{{{u 1}^{8}}\ {{u 4}^{4}}}-{{u 2}^{12}}}}, \:{xpnt = 1}\right] 
(5)
Type: List(Record(flg: Union("nil","sqfr","irred","prime"),fctr: Polynomial(Integer),xpnt: Integer))
fricas
q* t8_12_24 - B8_12_24

\label{eq6}0(6)
Type: Polynomial(Integer)




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