MathAction SandboxSomos2Eta

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- DedekindEtaFunctions
  - SandboxSomos2Eta <-- You are here.

Translation from Somos notation to eta-function notation and back

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

fricas

(1) -> <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

fricas

)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

fricas

)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>

fricas

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.

   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 12 OCT 2023 12:06:13 PM):

; wrote /var/aw/var/LatexWiki/POLYEVAL.NRLIB/POLYEVAL.fasl
; compilation finished in 0:00:00.012
------------------------------------------------------------------------
   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: 0.01 SEC.

   compiling local indexedSymbols : (String,List Integer) -> List Symbol
Time: 0.00 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: 0.04 SEC.

   compiling exported fromEta : (NonNegativeInteger,Polynomial Integer) -> Polynomial Integer
Time: 0.00 SEC.

(time taken in buildFunctor:  0)

;;;     ***       |SomosToEta| REDEFINED

;;;     ***       |SomosToEta| REDEFINED
Time: 0.00 SEC.

   Cumulative Statistics for Constructor SomosToEta
      Time: 0.06 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 12 OCT 2023 12:06:13 PM):

; wrote /var/aw/var/LatexWiki/STOETA.NRLIB/STOETA.fasl
; compilation finished in 0:00:00.024
------------------------------------------------------------------------
   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 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

$\label{eq5}\begin{array}{@{}l} \displaystyle \left[{\left[{flag = \verb#"prime"#}, \:{factor = q}, \:{exponent = 1}\right]}, \: \right. \ \ \displaystyle \left.{ \begin{array}{@{}l} \displaystyle \left[{flag = \verb#"prime"#}, \: \right. \ \ \displaystyle \left.{factor ={{8 \ q \ {{u 1}^{4}}\ {{u 2}^{2}}\ {{u 4}^{2}}\ {{u 8}^{4}}}+{{{u 1}^{8}}\ {{u 4}^{4}}}-{{u 2}^{12}}}}, \: \right. \ \ \displaystyle \left.{exponent = 1}\right]$

(5)

Type: List(Record(flag: Union("nil","sqfr","irred","prime"),factor: Polynomial(Integer),exponent: NonNegativeInteger?))

fricas

q* t8_12_24 - B8_12_24

$\label{eq6}0$

(6)

Type: Polynomial(Integer)

Subject: Be Bold !!

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