[om] Algebraic Numbers

[Andreas Strotmann]

Clare,

you may want to take a brief look at the old OpenMath Objectives paper 
for a reason why these algebraic numbers probably need to be used via 
bound variables (you sometimes have to make sure that you get the *same* 
root in all cases in an expression).

Clare So wrote:

>Dear OpenMathers,
>
>The MONET group at ORCCA is trying to figure out the appropriate
>OpenMath representation for algebraic numbers.  We propose two
>approaches.  Let's say we would like to encode $ {\alpha}{x^2} + 
>(\alpha - 1)x + (1 + \alpha )y $ in OpenMath where \alpha is
>the algebraic expression representing RootOf(z^2+z+1):
>
>* First Approach *
>
><OMA>
>  <OMS cd="algext1" name="algebraicnumber"/>
>  <OMBIND>
>    <OMS cd="fns1" name="lambda"/>
>    <OMBVAR>
>      <OMV name="alpha"/>
>    </OMBVAR>
>    <OMA>
>      [$ {\alpha ^ 2} + \alpha + 1 $ in OpenMath]
>    </OMA>
>    <OMA>
>      [$ {\alpha}{x^2} + (\alpha - 1)x + (1 + \alpha )y $ in OpenMath]
>    </OMA>
>  </OMBIND>
></OMA>
>
>  
>

This is technically not allowed (OMBIND only takes a single body 
expression), but I think it's on the right track. What you may need is 
something like this (using a much abbreviated syntax):

(1)  apply(binding(lambda,alpha, [alpha*x^2+alpha+1]),
       apply(algebraicnumber, binding(lambda,alpha,alpha^2+1))

Since algebraic numbers could in principle be defined as closed-form 
expressions involving nth-roots and complex integers, you may need to 
change this slightly to:

(2)  apply(binding(lambda,alpha, [alpha*x^2+alpha+1]),
        apply(algebraicnumber, 
apply(anyroot,binding(lambda,alpha,[alpha^2+1]),C))

(i.e. first construct a root over the complex numbers and then "cast" it 
to algebraic number).

Notice that this doesn't say what name the algebraic number is going to 
have, which is consistent with the way that indefinite integration is 
specified in OpenMath. 

An exact mirror of the integration symbol would allow you to say

(3)   ( < apply(algebraicnumber, binding(lambda,alpha,alpha^2+1) > 
)*x^2+alpha+1),

since the variable alpha would be implicitly exported, and thus alpha 
well-defined (although free) outside the lambda.  However, that may be 
asking a bit much, and one could consider re-writing the example again as

(4)   (apply(apply(algebraicnumber, 
binding(lambda,alpha,alpha^2+1)),alpha)*x^2+alpha+1),

with a signature of algebraicnmuber: unary-function -> unary-function, 
where the resulting unary function might be defined to be defined only 
if the argument is the algebraic number it represents (thus forcing the 
correct interpretation for the argument variable).

In all, though, I think the cleanest approach given the limited ability 
of Openmath to express co-references would be to ask that an OpenMath 
version of the statement "let alpha be the algebraic number defined by 
.." be wrapped around an expression that uses algebraic numbers, as in 
the first two examples (1) or (2) or via some equation.

This is essentially your approach below, except for noting that one can 
bind the algebraic number to a variable in order to avoid having to 
repeat the algebraic number's definition all the time, and except for 
making explicit the need to use a functional argument for 
"algebraicnumber" below to make explicit the variable with respect to 
which the root is taken.

 -- Andreas

>For this approach, alpha is defined in the second argument of of
><OMS cd="fns1" name="lambda"/> and is used in the expression in the
>third argument.
>
>
>* Second Approach *
>
><OMA>
>  <OMS cd ="arith1" name="plus"/>
>  <OMA>
>    <OMS cd="arith1" name="times"/>
>    <OMA>
>      <OMS cd="algext1" name="algebraicnumber"/>
>      [$ {\alpha ^ 2} + \alpha + 1 $ in OpenMath]
>    </OMA>
>    <OMA>
>      <OMS cd="arith1" name="power"/>
>      <OMV name="x"/>
>      <OMI> 2 </OMI>
>    </OMA>
>  </OMA>
>  <OMA>
>    <OMS cd="arith1" name="times"/>
>    <OMA>    
>      <OMS cd="arith1" name="minus"/>
>      <OMA>
>        <OMS cd="algext1" name="algebraicnumber"/>
>        [$ {\alpha ^ 2} + \alpha + 1 $ in OpenMath]
>      </OMA>
>      <OMI> 1 </OMI>
>    </OMA>
>    <OMV name="x"/>
>  </OMA>
>  <OMA>
>    <OMS cd="arith1" name="times"/>
>    <OMA>
>      <OMS cd="arith1" name="plus"/>
>      <OMA>
>        <OMS cd="algext1" name="algebraicnumber"/>
>        [$ {\alpha ^ 2} + \alpha + 1 $ in OpenMath]
>      </OMA>
>      <OMI> 1 </OMI>
>    </OMA>
>    <OMV name="y"/>
>  </OMA>
></OMA>
>
>The algebraic number is defined every time we need it.
>
>
>First approach is a compact way to express algebraic number because
>alpha is defined once for the entire expression.  Second approach is
>a straight-forward way because alpha is defined exactly the place we
>need it.
>
>Please let us know which approach you think should be used for this
>purpose.
>
>
>Clare & MONET group at UWO
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>  
>

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