[om] Algebraic Numbers

[Clare So]

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>

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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