OpenMath Content Dictionary: polyd2

Canonical URL:

http://www.openmath.org/cd/polyd2.ocd

CD Base:

http://www.openmath.org/cd

CD File:

polyd2.ocd

CD as XML Encoded OpenMath:

polyd2.omcd

Defines:

elimination, graded_lexicographic, graded_reverse_lexicographic, lexicographic, matrix_ordering, ordering, reverse_lexicographic, weighted, weighted_degree

Date:

2004-07-07

Version:

3

Review Date:

2006-04-01

Status:

experimental

---

This CD defines symbols for ordering of monomial for Distributed Multivariate Polynomials, which were defined in polyd1.

Original OpenMath v1.1 Poly 1997
Update to Current Format 1999-07-07 DPC
Move the names of rings to setname.ocd 1999-11-09 JHD
Delete those items moved to the new poly.ocd 1999-11-14 JHD
Delete those items pertaining to Groebner bases 2004-07-07 AMC

These are of use for canonical ways of writing polynomials and for Groebner bases

---

ordering

Description:

Used as an attribute to indicate an ordering of the monomials in a polynomial or list of polynomials. The value of this attribute should be one of the constructors specifying ordering.

Signatures:

sts

---

[Next: lexicographic] [Last: weighted_degree] [Top]

          The following orders on monomials have their standards definitions, 
         see, for example, "Ideals, Varieties and Algorithms", D. Cox, 
         J.B. Little and D. O'Shea, Springer Verlag.

---

lexicographic

Description:

The lexicographic ordering of monomials.

Signatures:

sts

---

[Next: reverse_lexicographic] [Previous: ordering] [Top]

---

reverse_lexicographic

Description:

The reverse lexicographic ordering of monomials

Signatures:

sts

---

[Next: graded_lexicographic] [Previous: lexicographic] [Top]

---

graded_lexicographic

Description:

Total degree order, graded with the lexicographic ordering.

Signatures:

sts

---

[Next: graded_reverse_lexicographic] [Previous: reverse_lexicographic] [Top]

---

graded_reverse_lexicographic

Description:

Total degree order, graded with the reverse lexicographic ordering.

Signatures:

sts

---

[Next: elimination] [Previous: graded_lexicographic] [Top]

---

elimination

Description:

This is an ordering, which is partially in terms of one ordering, and partially in terms of another. First argument is a number of variables. Second is ordering to apply on the first so many variables. Third is an ordering on the rest, to be used to break ties.

Example:

OpenMath XML (source)

         <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
           <OMA>
             <OMS name="elimination" cd="polyd2"/>
             <OMI> 1 </OMI>
             <OMS name="lexicographic" cd="polyd2"/>
             <OMS name="graded_reverse_lexicographic" cd="polyd2"/>
           </OMA>
         </OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="polyd2">elimination</csymbol>
  <cn type="integer">1</cn>
  <csymbol cd="polyd2">lexicographic</csymbol>
  <csymbol cd="polyd2">graded_reverse_lexicographic</csymbol>
 </apply>
</math>

Prefix

elimination ( 1 , lexicographic, graded_reverse_lexicographic)

Popcorn

polyd2.elimination(1, polyd2.lexicographic, polyd2.graded_reverse_lexicographic)

Rendered Presentation MathML

$$\mathrm{elimination}(1,\mathrm{lexicographic},\mathrm{graded\_reverse\_lexicographic})$$

Signatures:

sts

---

[Next: matrix_ordering] [Previous: graded_reverse_lexicographic] [Top]

---

matrix_ordering

Description:

The argument is a matrix with as many columns as indeterminates (= rank). Each row in turm is multiplied by the column vector of exponents to produce a weighting for comparison purposes.

Signatures:

sts

---

[Next: weighted] [Previous: elimination] [Top]

---

weighted

Description:

The first argument is a list of integers to act as variable weights, and the second is an ordering. The result is an ordering.

Signatures:

sts

---

[Next: weighted_degree] [Previous: matrix_ordering] [Top]

  We need a few more orderings... 

     Definition of some other constructors

---

weighted_degree

Description:

The total degree of its argument, taking into account any weights declared. The value returned is an integer: non-negative if the weights are. We note that the degree of 0 is undefined.

Example:

OpenMath XML (source)

        <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
          <OMA>
            <OMS name="eq" cd="relation1"/>
            <OMA>
              <OMS name="weighted_degree" cd="polyd2"/>
              <OMA>
                <OMS name="DMP" cd="polyd1"/>
                <OMA>
                  <OMS name="poly_ring_d" cd="polyd1"/>
                  <OMS name="Q" cd="setname1"/>
                  <OMI> 3 </OMI>
                </OMA>
                <OMATTR>
                  <OMATP>
                    <OMS name="ordering" cd="polyd2"/>
                    <OMA>
                      <OMS name="weighted" cd="polyd2"/>
                      <OMA>
                        <OMS name="list" cd="list1"/>
                        <OMI> 1 </OMI>
                        <OMI> 2 </OMI>
                        <OMI> 3 </OMI>
                      </OMA>
                      <OMS name="graded_lexicographic" cd="polyd2"/>
                    </OMA>
                  </OMATP>
                  <OMA>
                    <OMS name="SDMP" cd="polyd1"/>
                    <OMA>
                      <OMS name="term" cd="polyd1"/>
                      <OMI> 1 </OMI>
                      <OMI> 0 </OMI>
                      <OMI> 0 </OMI>
                      <OMI> 1 </OMI>
                    </OMA>
                    <OMA>
                      <OMS name="term" cd="polyd1"/>
                      <OMI> 2 </OMI>
                      <OMI> 2 </OMI>
                      <OMI> 0 </OMI>
                      <OMI> 0 </OMI>
                    </OMA>
                    <OMA>
                      <OMS name="term" cd="polyd1"/>
                      <OMI> 3 </OMI>
                      <OMI> 0 </OMI>
                      <OMI> 1 </OMI>
                      <OMI> 0 </OMI>
                    </OMA>
                    <OMA>
                      <OMS name="term" cd="polyd1"/>
                      <OMI> 4 </OMI>
                      <OMI> 1 </OMI>
                      <OMI> 0 </OMI>
                      <OMI> 0 </OMI>
                    </OMA>
                  </OMA>
                </OMATTR>
              </OMA>
            </OMA>
            <OMI> 3 </OMI>
          </OMA>
        </OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="polyd2">weighted_degree</csymbol>
   <apply><csymbol cd="polyd1">DMP</csymbol>
    <apply><csymbol cd="polyd1">poly_ring_d</csymbol>
     <csymbol cd="setname1">Q</csymbol>
     <cn type="integer">3</cn>
    </apply>
    <semantics>
     <apply><csymbol cd="polyd1">SDMP</csymbol>
      <apply><csymbol cd="polyd1">term</csymbol>
       <cn type="integer">1</cn>
       <cn type="integer">0</cn>
       <cn type="integer">0</cn>
       <cn type="integer">1</cn>
      </apply>
      <apply><csymbol cd="polyd1">term</csymbol>
       <cn type="integer">2</cn>
       <cn type="integer">2</cn>
       <cn type="integer">0</cn>
       <cn type="integer">0</cn>
      </apply>
      <apply><csymbol cd="polyd1">term</csymbol>
       <cn type="integer">3</cn>
       <cn type="integer">0</cn>
       <cn type="integer">1</cn>
       <cn type="integer">0</cn>
      </apply>
      <apply><csymbol cd="polyd1">term</csymbol>
       <cn type="integer">4</cn>
       <cn type="integer">1</cn>
       <cn type="integer">0</cn>
       <cn type="integer">0</cn>
      </apply>
     </apply>
     <annotation-xml cd="polyd2" name="ordering">
      <apply><csymbol cd="polyd2">weighted</csymbol>
       <apply><csymbol cd="list1">list</csymbol>
        <cn type="integer">1</cn>
        <cn type="integer">2</cn>
        <cn type="integer">3</cn>
       </apply>
       <csymbol cd="polyd2">graded_lexicographic</csymbol>
      </apply>
     </annotation-xml>
    </semantics>
   </apply>
  </apply>
  <cn type="integer">3</cn>
 </apply>
</math>

Prefix

eq (weighted_degree (DMP (poly_ring_d (Q, 3 ) , Attrib([ ordering weighted (list ( 1 , 2 , 3 ) , graded_lexicographic) ], SDMP (term ( 1 , 0 , 0 , 1 ) , term ( 2 , 2 , 0 , 0 ) , term ( 3 , 0 , 1 , 0 ) , term ( 4 , 1 , 0 , 0 ) ) ) ) ) , 3 )

Popcorn

polyd2.weighted_degree(polyd1.DMP(polyd1.poly_ring_d(setname1.Q, 3), polyd1.SDMP(polyd1.term(1, 0, 0, 1), polyd1.term(2, 2, 0, 0), polyd1.term(3, 0, 1, 0), polyd1.term(4, 1, 0, 0)){polyd2.ordering -> polyd2.weighted([1 , 2 , 3], polyd2.graded_lexicographic)})) = 3

Rendered Presentation MathML

$$\mathrm{weighted\_degree}\left(\mathrm{DMP}(\mathrm{poly\_ring\_d}(\mathbb{Q},3),\mathrm{SDMP}(\mathrm{term}(1,0,0,1),\mathrm{term}(2,2,0,0),\mathrm{term}(3,0,1,0),\mathrm{term}(4,1,0,0)))\right)=3$$

Signatures:

sts

---

[First: ordering] [Previous: weighted] [Top]