JAS project users guide

# Interactive scripting guide

This document contains first information on how-to use the interactive scripting of the JAS project. It can be used via the Java Python interpreter `jython`, or the Java Ruby interpreter `jruby` or the jruby Android App `Ruboto-IRB`.
The usage of JAS as an ordinary Java library, adding `jas.jar` to the classpath and creating and using objects from JAS classes, is introduced in the API guide.

JAS can be started with the script "`jas`" in the JAS home directory. By default the JRuby interactive shell ist used. For the Jython shell use "`jas -py`". When started from a desktop, like MathLibre, the shells will look as in the following picture. The upper right terminal shows a Jython shell and the lower left terminal shows a JRuby shell.

JAS jython and jruby interface in MathLibre

### Getting started

As first example we will discus how to compute a Groebner base with `jruby`. The jruby script will be placed into a file, e.g. `getstart-gb.rb`. This script file is executed by calling

      jruby getstart-gb.rb

If you start `jruby` (or `jas -rb`) without a file name, then an interactive shell is opened and you can type commands and expressions as desired. The script file first imports the desired mathematical classes from the `jas.rb` script which does all interfacing to the Java library. For the Rdoc of it see here.

      require "examples/jas"

In our case we need `PolyRing` to define an appropriate polynomial ring and later `Ideal` to define sets of polynomials and have methods to compute Groebner bases. `PolyRing` takes arguments for required definitions of the polynomial ring: the type of the coefficient ring, the names of the used variables and the desired term order.

      r = PolyRing.new( QQ(), "B,S,T,Z,P,W", PolyRing.lex)

The ring definition is stored in the variable `r` for later use. The string `"QQ()"` defines the coefficient ring to be the rational numbers, the polynomial ring consists of the variables `B, S, T, Z, P, W` and the term order `PolyRing.lex` means a lexicographic term order. For some historical reason the term order orders the variables as `B < S < T < Z < P < W` and not the other way. I.e. the highest or largest variable is always on the right of the list of variables not on the left as in some other algebra systems. With

      puts "PolyRing: " + r.to_s

you can print out the ring definition. `r.to_s` is the usual Ruby way of producing string representations of objects, which in our case calls the respective Java method `toScript()` of the JAS object. It produces

      PolyRing: PolyRing.new(QQ(),"B,S,T,Z,P,W",PolyRing.lex)

i.e. the same expression as defined above. In general the string from `r.to_s` of an JAS object can be used via cut-and-past as new input. Next we need to enter the generating polynomials for the ideal. We do this in three steps, first define the Ruby variables for the polynomial ring, next define the polynomials and then the creation of the ideal using the ring definition from before and the polynomial list.

      one,B,S,T,Z,P,W = r.gens()

Small letter variables for polynomials are defined automatically but because of Ruby handling capital letter variables as constant they must be defined by hand. The method `r.gens()` returns a list of all generators (variables and values) of the polynomial ring.

    ff = [
     45 * P + 35 * S - 165 * B - 36,
     35 * P + 40 * Z + 25 * T - 27 * S,
     15 * W + 25 * S * P + 30 * Z - 18 * T - 165 * B**2,
     - 9 * W + 15 * T * P + 20 * S * Z,
     P * W + 2 * T * Z - 11 * B**3,
     99 * W - 11 * B * S + 3 * B**2,
     B**2 + 33/50 * B + 2673/10000
    ];

The polynomial list can be generated by any means Ruby allows for polynomial expressions. In our example we use Ruby brackets `[ ... ]` for the creation of the list. The polynomials in the list are delimited by commas, and may be enclosed in parentheses. The syntax for polynomials is the Ruby expression syntax including literals from the coefficient ring `QQ()`, variables and operators `+, -, *, **` (for summation, subtraction, multiplication, and exponentiation). The ideal is then defined with

      f = r.ideal( "", ff )

It is contained the the polynomial ring `r` by construction and consists of the polynomials from the list `ff`, the first parameter is the empty string. Ideals can be printed with

      puts "Ideal: " + f.to_s

In this example it produces the following output.

    Ideal: SimIdeal.new(PolyRing.new(QQ(),"B,S,T,Z,P,W",PolyRing.lex),
           "",[( B**2 + 33/50 * B + 2673/10000 ),
               ( 45 * P + 35 * S - 165 * B - 36 ),
               ( 35 * P + 40 * Z + 25 * T - 27 * S ),
               ( 15 * W + 25 * S * P + 30 * Z - 18 * T - 165 * B**2 ),
               ( ( -9 ) * W + 15 * T * P + 20 * S * Z ),
               ( 99 * W - 11 * B * S + 3 * B**2 ),
               ( P * W + 2 * T * Z - 11 * B**3 )])

The polynomial terms are now sorted with respect to the lexicographical term order. The highest term is first in a polynomial. Also the polynomials are sorted with respect to the term order, but with smallest polynomial first in the list. Finaly we can go to the computation of the Groebner basis of this ideal.

      g = f.GB()

The ideal `f` has a method `GB()` which computes the Groebner base. The computed Groebner base is stored in the variable `g` which is also an ideal. It can be printed in the same was as the ideal `f`

      puts "Groebner base: " + g.to_s

The output first shows the output from calling the `GB()` method and the the ideal basis.

    sequential(field) GB executed in 37 ms

    Groebner base: SimIdeal.new(PolyRing.new(QQ(),"B,S,T,Z,P,W",PolyRing.lex),
                   "",[( B**2 + 33/50 * B + 2673/10000 ),
                       ( S - 5/2 * B - 9/200 ),
                       ( T - 37/15 * B + 27/250 ),
                       ( Z + 49/36 * B + 1143/2000 ),
                       ( P - 31/18 * B - 153/200 ),
                       ( W + 19/120 * B + 1323/20000 )])

The Groebner base was computed with the sequential algorithm or polynomial rings over fields in 37 ms and consists of six polynomials. The polynomials are now monic, i.e. the leading coefficient is 1 and omitted during print out. This concludes the first getting started section.

### Overview of Ruby and Python classes and methods in jas.rb and jas.py

The jruby and the jython interface to the JAS library constain the following classes. The class and method names are almost identical, except where name clashes with scripting language occur, e.g. `Ideal` in jython, but `SimIdeal` in jruby. The class constructors in Ruby are used with the `.new()` method and in Python the class name is use like a function name. For example the construction of a polynomial ring is done in Ruby by `PolyRing.new(...)` and in Python by `PolyRing(...)`. For the Rdoc of them see here and for the Epydoc of them see here.

- `PolyRing`, `Ideal`/`SimIdeal` and `ParamIdeal`
  define polynomial rings, ideals and ideals over rings with coefficient parameters.
  `Ideal` has methods for sequential, parallel and distributed Groebner bases computation, for example `GB()`, `isGB()`, `parGB()`, `distGB()`, `NF()` and `intersect()`.
  `ParamIdeal` has methods for comprehensive Groebner bases computation, for example `CGB()`, `CGBsystem()`, `regularGB()`,

- `SolvPolyRing` and `SolvableIdeal`/`SolvIdeal`
  define solvable polynomial rings and left, right and two-sided ideals.
  `SolvableIdeal` has methods for left, right and two-sided Groebner bases computation, e.g. `leftGB()`, `rightGB()`, `twosidedGB()`, `intersect()`.

- `Module` and `SubModule`
  define modules over polynomial rings and sub modules.
  `Module` has a method for sequential Groebner bases computation, e.g. `GB()`.

- `SolvableModule` and `SolvableSubModule`
  define modules over solvable polynomial rings and sub modules.
  `SolvableModule` has methods for left, right and two-sided Groebner bases computation, e.g. `leftGB()`, `rightGB()`, `twosidedGB()`.

Ruby has support for rational numbers, so a literal, like `2/3`, is recognized as rational number 2/3. Python has no support for rational number literals and `2/3` is recognized as interger division, resulting in the integer `0` (zero). To allow rational numbers in JAS, the Python tuple or list notation must be used to express rational numbers, so `(2,3)` is recognized as rational number 2/3.

For example in the construction of Legendre polynomials a rational number `r = 1/n` appears. As tuple literal it is written `(1,n)` and as list literal it can be written as `[1,n]`.

      p = (2*n-1) * x * P[n-1] - (n-1) * P[n-2];
      r = (1,n); # no rational numbers in Python, use tuple notation
      p = r * p;

In the same way complex rational numbers can be written as nested tuples. For example `1/n + 1/2 i` can be written as `((1,n),(1,2))`. If the second list element is omited it is asumed to be one as rational number and zero as complex number. To avoid ambiguities use a trailing comma, as in `((1,2),)`.

### Overview of some mathematical capabilities of JAS

In this section we summarize some mathematical constructions which are possible with JAS: real root computation, power series and non-commutative polynomial rings.

#### Real roots of zero dimensional ideals

Besides the computation of Gröbner bases JAS is able to use them to solve various other problems. In this sub-section we present the computation of real roots of systems of (algebraic) equations. When the system of equations has only finitely many real roots, such systems define so called zero dimensional ideals, they can be computed (using jython) as follows.

      r = PolyRing(QQ(),"x,y,z",PolyRing.lex);
      print "Ring: " + str(r);
      print;

      [one,x,y,z] = r.gens(); # is also automatic

      f1 = (x**2 - 5)*(x**2 - 3)**2;
      f2 = y**2 - 3;
      f3 = z**3 - x * y;

      F = r.ideal( list=[f1,f2,f3] );

      R = F.realRoots();
      F.realRootsPrint()

In the above example we compute the real roots of the equations defined by the polynomials `f1, f2, f3`. First we define the polynomial ring and then construct the ideal `F` from the given polynomials. The method `F.realRoots()` computes the real roots and method `F.realRootsPrint()` prints a decimal approximation of tuples of real roots. The output of the last method call looks as follows.

    [-1.7320508076809346675872802734375, -1.7320508076809346675872802734375, 1.4422495705075562000274658203125]
    [1.7320508076809346675872802734375, 1.7320508076809346675872802734375, 1.4422495705075562000274658203125]

    [1.7320508076809346675872802734375, -1.7320508076809346675872802734375, -1.4422495705075562000274658203125]
    [-1.7320508076809346675872802734375, 1.7320508076809346675872802734375, -1.4422495705075562000274658203125]

    [0.50401716955821029841899871826171875, 2.236067977384664118289947509765625, -1.7320508076809346675872802734375, -1.5704178023152053356170654296875]
    [-0.50401716955821029841899871826171875, -2.236067977384664118289947509765625, 1.7320508076809346675872802734375, -1.5704178023152053356170654296875]
    [-3.96811878503649495542049407958984375, -2.236067977384664118289947509765625, -1.7320508076809346675872802734375, 1.5704178023152053356170654296875]
    [3.96811878503649495542049407958984375, 2.236067977384664118289947509765625, 1.7320508076809346675872802734375, 1.5704178023152053356170654296875]

The roots in the tuples `[-1.732..., -1.732..., 1.442...]` correspond to the roots in the variables `[x, y, z]`. The last four tuples have four entries `[0.504..., 2.236..., -1.732..., -1.570...]`, where the first entry stems from an internal field extension, which was needed to correctly identify the roots of the ideal and are to be ignored. That is the tuple `[2.236..., -1.732..., -1.570...]` without the first entry is a real root of the ideal. That is, the decimal approximation of the real roots are the following 8 tuples.

      [-1.73205..., -1.73205...,  1.44224...]
      [ 1.73205...,  1.73205...,  1.44224...]

      [ 1.73205..., -1.73205..., -1.44224...]
      [-1.73205...,  1.73205..., -1.44224...]

      [ 2.23606..., -1.73205..., -1.57041...]
      [-2.23606...,  1.73205..., -1.57041...]
      [-2.23606..., -1.73205...,  1.57041...]
      [ 2.23606...,  1.73205...,  1.57041...]

More details and further examples can be found in the jython file `0dim_real_roots.py`.

#### Power series

Univariate power series can be constructed via the `SeriesRing` class and an multivariate power series with the `MultiSeriesRing` class. There are short cut methods `PS(coeff, name, truncate, function)` and `MPS(coeff, names, truncate, function)` to construct a power series with a given coefficient generator '`function`'. In the following example (using jython) we create a new power series ring `pr` in the variable `y` over the rational numbers. The creation of power series is done in the same way as polynomials are created. There are additional methods like `r.exp()` or `r.sin()` to create the exponential power series or the power series for the sinus function.

      pr = SeriesRing("Q(y)");
      print "pr:", pr;

      one = pr.one();
      r1 = pr.random(4);
      r2 = pr.random(4);

      print "one:", one;
      print "r1:", r1;
      print "r2:", r2;

      r4 = r1 * r2 + one;
      e = pr.exp();
      r5 = r1 * r2 + e;

      print "e:", e;
      print "r4:", r4;
      print "r5:", r5;

Once power series are created, for example `r1, r2, e` above, it is possible to use arithmetic operators to built expressions of power series like '`r1 * r2 + one`' or '`r1 * r2 + e`'.

    pr: PS(QQ(),"y",11)

    one: 1
    r1:  (13,5) - (14,5) * y**3 - y**4 + 14 * y**5 - (12,7) * y**6 - 4 * y**7 - (9,14) * y**8 + 3 * y**9 + (1,15) * y**10

    r2:  - (9,16) * y + (5,6) * y**3 + (2,3) * y**5 + (5,6) * y**9 + (5,2) * y**10

    e: 1 + y + (1,2) * y**2 + (1,6) * y**3 + (1,24) * y**4 + (1,120) * y**5 + (1,720) * y**6 + (1,5040) * y**7 + (1,40320) * y**8
       + (1,362880) * y**9 + (1,3628800) * y**10

    r4: 1 - (117,80) * y + (13,6) * y**3 + (63,40) * y**4 + (551,240) * y**5 - (245,24) * y**6 + (11,84) * y**7 + (241,20) * y**8 + (97,224) * y**9 + (173,16) * y**10

    r5: 1 - (37,80) * y + (1,2) * y**2 + (7,3) * y**3 + (97,60) * y**4 + (553,240) * y**5 - (7349,720) * y**6 + (661,5040) * y**7 + (485857,40320) * y**8
        + (157141,362880) * y**9 + (39236401,3628800) * y**10

It is also possible to create power series by defining a generating function or by defining a fixed point with respect to a map between power series.

      def g(a):
          return a+a;
      ps1 = pr.create(g);

      class coeff( Coefficients ):
          def generate(self,i):
              ...
      ps6 = pr.create( clazz=coeff( pr.ring.coFac ) );

      class cosmap( PowerSeriesMap ):
          def map(self,ps):
              ...
      ps8 = pr.fixPoint( cosmap( pr.ring.coFac ) );

More details and further examples can be found in the jython file `powerseries.py` and `powerseries_multi.py` and their Ruby counter parts.

#### Solvable polynomial rings

Solvable polynomial rings are non commutative polynomial rings where the non commutativity is expressed by commutator relations. Commutator relations are stored in a data structure called relation table. In the definition of a solvable polynomial ring this relation table must be defined. E.g the definition for the ring of a solvable polynomial ring (in jruby) is

      require "examples/jas"
      # WA_32 solvable polynomial example

      p = PolyRing.new(QQ(),"a,b,e1,e2,e3");
      relations = [e3, e1, e1*e3 - e1,
                   e3, e2, e2*e3 - e2];

      puts "relations: = " + relations.join(", ") { |r| r.to_s };

    relations: = e3, e1, ( e1 * e3 - e1 ), e3, e2, ( e2 * e3 - e2 )

The relation table must be build from triples of (commutative) polynomials. A triple `p1, p2, p3` is interpreted as non commutative multiplication relation `p1 .*. p2 = p3`. `p1` and `p2` must be a single term, single variable polynomials. The term order must be choosen such that leadingTerm(`p1 p2`) equals leadingTerm(`p3`) and `p1 > p2` for each triple. The polynomial `p3` is in commutative form, i.e. multiplication operators occuring in it are commutative. Variables for which there are no commutator relations are assumed to commute with each other and with all other variables, e.g. the variables `a, b` in the example.

      rp = SolvPolyRing.new(QQ(), "a,b,e1,e2,e3", PolyRing.lex, relations);
      puts "SolvPolyRing: " + rp.to_s;

      puts "gens = " + rp.gens().join(", ") { |r| r.to_s };
      one,a,b,e1,e2,e3 = rp.gens();

      f1 = e1 * e3**3 + e2**10 - a;
      f2 = e1**3 * e2**2 + e3;
      f3 = e3**3 + e3**2 - b;

      F = [ f1, f2, f3 ];
      puts "F = " + F.join(", ") { |r| r.to_s };

      I = rp.ideal( "", F );
      puts "SolvableIdeal: " + I.to_s;

After the definition of the variables `e1, e2, e3` as non-commutative as elements of the ring `rp`, the expressions for the polynomials `f1, f2, f3` use non-cummutative multiplication with the `*` operator.

A complete example is contained in the jRuby script `solvablepolynomial.rb`. Running the script computes a left, right and twosided Groebner base for the following ideal `I` generated by the polynomial list `F`.

    ring is associative
    SolvPolyRing: SolvPolyRing.new(QQ(),"a,b,e1,e2,e3",PolyRing.lex,rel=[e3, e2, ( e2 * e3 - e2 ), e3, e1, ( e1 * e3 - e1 )])

    gens = 1, a, b, e1, e2, e3
    F = ( e1 * e3**3 + e2**10 - a ), ( e3 + e1**3 * e2**2 ), ( e3**3 + e3**2 - b )

    SolvableIdeal: SolvIdeal.new(SolvPolyRing.new(QQ(),"a,b,e1,e2,e3",PolyRing.lex,
                          rel=[e3, e2,  ( e2 * e3 - e2 ), e3, e1,  ( e1 * e3 - e1 )]),
                      "",[( e3 + e1**3 * e2**2 ), ( e3**3 + e3**2 - b ), ( e1 * e3**3 + e2**10 - a )])

The left Groebner base is

    sequential(field|nocom) leftGB executed in 29 ms

    seq left GB: SolvIdeal.new(SolvPolyRing.new(QQ(),"a,b,e1,e2,e3",PolyRing.lex,rel=[e3, e2,  ( e2 * e3 - e2 ), e3, e1,  ( e1 * e3 - e1 )]),
                 "",[a, b, e1**3 * e2**2, e2**10, e3])

the twosided Groebner base is

    sequential(field|nocom) twosidedGB executed in 28 ms
    seq twosided GB: SolvIdeal.new(SolvPolyRing.new(QQ(),"a,b,e1,e2,e3",PolyRing.lex,rel=[e3, e2,  ( e2 * e3 - e2 ), e3, e1,  ( e1 * e3 - e1 )]),
                     "",[a, b, e1, e2, e3])

and the right Groebner base is

    sequential(field|nocom) rightGB executed in 16 ms
    seq right GB: SolvIdeal.new(SolvPolyRing.new(QQ(),"a,b,e1,e2,e3",PolyRing.lex,rel=[e3, e2,  ( e2 * e3 - e2 ), e3, e1,  ( e1 * e3 - e1 )]),
                  "",[a, b, e1, e2**10, e3])

### Using the internal polynomial parser

The internal polynomial parser has a simpler syntax than the Ruby or Python expression syntax. For example the multiplication operator `*` can be omitted and `^` can be used for exponentiation `**`. Moreover, `2/3` will work for rational numbers also in Python.

An example using the internal polynomial parser will be discused in the following. The jython script is be placed into a file, e.g. `getstart.py`. This script file is executed by calling

      jython getstart.py

If you start `jython` (or `jas -py`) without a file name, then an interactive shell is opened and you can type commands and expressions as desired. The script file first imports the desired mathematical classes from the `jas.py` script which does all interfacing to the Java library. For the Epydoc of it see here.

      from jas import Ring, Ideal

In our case we need `Ring` to define an appropriate polynomial ring and `Ideal` to define sets of polynomials and have methods to compute Groebner bases. `Ring` takes a string argument which contains required definitions of the polynomial ring: the type of the coefficient ring, the names of the used variables and the desired term order.

      r = Ring( "Rat(B,S,T,Z,P,W) L" );

The ring definition is stored in the variable `r` for later use. The string `"Rat(B,S,T,Z,P,W) L"` defines the coefficient ring to be the rational numbers `Rat`, the polynomial ring consists of the variables `B, S, T, Z, P, W` and the term order `L` means a lexicographic term order. For some historical reason the term order orders the variables as `B < S < T < Z < P < W` and not the other way. I.e. the highest or largest variable is always on the right of the list of variables not on the left as in some other algebra systems. With

      print "Ring: " + str(r);

you can print out the ring definition. `str(r)` is the usual Python way of producing string representations of objects, which in our case calls the respective Java method `toString()` of the JAS ring object. It produces

    Ring: BigRational(B, S, T, Z, P, W) INVLEX

i.e. the coefficients are from the jas class `BigRational` and the term order is `INVLEX` (`INV` because the largest variable is on the right). Next we need to enter the generating polynomials for the ideal. We do this in two steps, first define a Python string with the polynomials and then the creation of the ideal using the ring definition from before and the polynomial string.

    ps = """
    (
     ( 45 P + 35 S - 165 B - 36 ),
     ( 35 P + 40 Z + 25 T - 27 S ),
     ( 15 W + 25 S P + 30 Z - 18 T - 165 B**2 ),
     ( - 9 W + 15 T P + 20 S Z ),
     ( P W + 2 T Z - 11 B**3 ),
     ( 99 W - 11 B S + 3 B**2 ),
     ( B**2 + 33/50 B + 2673/10000 )
    )
    """;

The polynomial string can be generated by any means Python allows for string manipulation. In our example we use Python multiline strings, which are delimited by triple quotes `""" ... """`. The list of polynomials is delimited by parenthesis `( ... )`, as well as every polynomial is delimited by parenthesis, e.g. `( B**2 + 33/50 B + 2673/10000 )`. The polynomials are separated by commas. The syntax for polynomials is a sequence of monimals consisting of coefficients and terms (as products of powers of variables). The terms can optionally be written with multiplication sign, i.e. `25 S P` can be written `25*S*P`. Variable names must be delimited by white space or some operator, i.e. you can not write `25 SP` because `SP` is not a listed variable name in the polynomial ring definition. Coefficients may not contain white space, i.e. the `/` separating the nominator from the denominator may not be surrounded by spaces, i.e. writing `33 / 50` is not allowed. Powers of variables can be written with `**` or `^`, i.e. the square of `B` is written as `B**2` or `B^2`. The ideal is the defined with

      f = Ideal( r, ps );

The ideal is contained the the polynomial ring `r` and consists of the polynomials from the string `ps`. Ideals can be printed with

      print "Ideal: " + str(f);

In this example it produces the following output.

    Ideal: BigRational(B, S, T, Z, P, W) INVLEX
    (
    ( B^2 + 33/50 B + 2673/10000  ),
    ( 45 P + 35 S - 165 B - 36  ),
    ( 35 P + 40 Z + 25 T - 27 S ),
    ( 15 W + 25 S * P + 30 Z - 18 T - 165 B^2 ),
    ( -9 W + 15 T * P + 20 S * Z ),
    ( 99 W - 11 B * S + 3 B^2 ),
    ( P * W + 2 T * Z - 11 B^3 )
    )

The polynomial terms are now sorted with respect to the lexicographical term order. The highest term is first in a polynomial. Also the polynomials are sorted with respect to the term order, but with smallest polynomial first in the list. Finaly we can go to the computation of the Groebner basis of this ideal.

      g = f.GB();

The ideal `f` has a method `GB()` which computes the Groebner base. The computed Groebner base is stored in the variable `g` which is also an ideal. It can be printed in the same way as the ideal `f`

      print "Groebner base:", g;

The output first shows the output from calling the `GB()` method and the the ideal basis.

    sequential executed in 136 ms

    Groebner base: BigRational(B, S, T, Z, P, W) INVLEX
    (
    ( B^2 + 33/50 B + 2673/10000  ),
    ( S - 5/2 B - 9/200  ),
    ( T - 37/15 B + 27/250  ),
    ( Z + 49/36 B + 1143/2000  ),
    ( P - 31/18 B - 153/200  ),
    ( W + 19/120 B + 1323/20000  )
    )

I.e. the Groebner base was computed in 135 ms and consists of six polynomials. The polynomials are now monic, i.e. the leading coefficient is 1 and omitted during print out. This concludes the getting started section.

#### Solvable polynomial rings and the internal parser

Solvable polynomial rings are non commutative polynomial rings where the non commutativity is expressed by commutator relations. Commutator relations are stored in a data structure called relation table. In the definition of a solvable polynomial ring this relation table must be defined. E.g the definition for the ring of a solvable polynomial ring is

    Rat(a,b,e1,e2,e3) L
    RelationTable
    (
     ( e3 ), ( e1 ), ( e1 e3 - e1 ),
     ( e3 ), ( e2 ), ( e2 e3 - e2 )
    )

The relation table must be build from triples of (commutative) polynomials. A triple `p1, p2, p3` is interpreted as non commutative multiplication relation `p1 .*. p2 = p3`. Currently `p1` and `p2` must be single term, single variable polynomials. The term order must be choosen such that leadingTerm(`p1 p2`) equals leadingTerm(`p3`) and `p1 > p2` for each triple. Polynomial `p3` must be in commutative form, i.e. multiplication operators occuring in it are commutative. Variables for which there are no commutator relations are assumed to commute with each other and with all other variables, e.g. the variables `a, b` in the example. Polynomials in the generating set of an ideal are also assumed to be in commutative form. If you need non-commutative multiplication in the polynomial expresions, please use the jython or jruby interface, as discussed above.

A complete example is contained in the Python script `solvable.py`. Running the script computes a left, right and twosided Groebner base for the following ideal

    (
     ( e1 e3^3 + e2^10 - a ),
     ( e1^3 e2^2 + e3 ),
     ( e3^3 + e3^2 - b )
    )

The left Groebner base is

    (
     ( a ), ( b ),
     ( e1^3 * e2^2 ), ( e2^10 ), ( e3 )
    )

the twosided Groebner base is

    (
     ( a ), ( b ), ( e1 ), ( e2 ), ( e3 )
    )

and the right Groebner base is

    (
     ( a ), ( b ), ( e1 ), ( e2^10 ), ( e3 )
    )

A module example is in `armbruster.py` and a solvable module example is in `solvablemodule.py`.

### Overview of JAS Android App

The App has been developed for JAS version 2.5 on Android 5 and is not working on current Android versions (since 2019).

The JAS application uses the Ruboto-IRB Android application. Ruboto provides an jruby scripting interpreter together with an editor application. The Ruboto App is enhanced with the JAS jruby interface and the JAS Java classes.

For the Android app the main screen with the "trinks.rb" example and its output looks as follows.

The JAS jruby interface on Android has the same functionality as the general JAS jruby scripting interface (only some functionality of the power series is not avaliable).

Last modified: Mon Feb 28 10:39:28 CET 2022