OpenMath Content Dictionary: polynomial1

Canonical URL:

http://www.openmath.org/cd/polynomial1.ocd

CD Base:

http://www.openmath.org/cd

CD File:

polynomial1.ocd

CD as XML Encoded OpenMath:

polynomial1.omcd

Defines:

expand, coefficient, coefficient_ring, degree, leading_coefficient, leading_monomial, leading_term

Date:

2004-07-12

Version:

0

Review Date:

2006-07-12

Status:

experimental

Uses CD:

alg1, arith1, logic1, quant1, set1, setname1, setname2, relation1, fns1, interval1, integer1

This CD holds a collection of basic constructs for univariate polynomials over rings. The data structures for polynomials can be arithmetic expressions, for instance using the ring1.expression symbol, or DMP as in the CD polyd1.

coefficient_ring

Role:

application

Description:

This symbol is a unary function whose argument should be a polynomial. It represents the coefficient ring of the polynomial.

Example:

The coefficient ring is often explicitly given as part of the polynomial:

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMA><OMS cd="relation1" name="eq"/>
     <OMA><OMS cd="polynomial1" name="coefficient_ring"/>
          <OMA><OMS cd="ring1" name="expression"/>
               <OMA><OMS cd="ring1" name="expression"/>
                    <OMA><OMS cd="ring3" name="poly_ring"/>
                         <OMA><OMS cd="ringname1" name="Zm"/>
                              <OMI>7</OMI>
                         </OMA>
                         <OMV name="X"/>
                    </OMA>
                    <OMA><OMS cd="arith1" name="power"/>
                         <OMA><OMS cd="arith1" name="plus"/>
                              <OMV name="X"/><OMI>1</OMI>
                         </OMA>
                         <OMI>2</OMI>
                    </OMA>
               </OMA>
          </OMA>
     </OMA>
     <OMA><OMS cd="ringname1" name="Zm"/>
          <OMI>7</OMI>
     </OMA>
</OMA>
  </OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="polynomial1">coefficient_ring</csymbol>
   <apply><csymbol cd="ring1">expression</csymbol>
    <apply><csymbol cd="ring1">expression</csymbol>
     <apply><csymbol cd="ring3">poly_ring</csymbol>
      <apply><csymbol cd="ringname1">Zm</csymbol><cn>7</cn></apply>
      <ci>X</ci>
     </apply>
     <apply><csymbol cd="arith1">power</csymbol>
      <apply><csymbol cd="arith1">plus</csymbol><ci>X</ci><cn>1</cn></apply>
      <cn>2</cn>
     </apply>
    </apply>
   </apply>
  </apply>
  <apply><csymbol cd="ringname1">Zm</csymbol><cn>7</cn></apply>
 </apply>
</math>

Prefix

eq (coefficient_ring (expression (expression (poly_ring (Zm (7) , X) , power (plus ( X, 1) , 2) ) ) ) , Zm (7) )

Popcorn

polynomial1.coefficient_ring(ring1.expression(ring1.expression(ring3.poly_ring(ringname1.Zm(7), $X), ($X + 1) ^ 2))) = ringname1.Zm(7)

Rendered Presentation MathML

$$\mathrm{coefficient\_ring}\left(\mathrm{expression}\left(\mathrm{expression}(\mathrm{poly\_ring}(\mathrm{Zm}\left(7\right),X),{(X+1)}^2)\right)\right)=\mathrm{Zm}\left(7\right)$$

Signatures:

sts

coefficient

Role:

application

Description:

This symbol is a binary function whose first argument should be a polynomial f and whose second argument should be a non-negative integer n. It represents the coefficient of the i-th power of the variable in the polynomial f.

Example:

The coefficient of the first power of X in the polynomial (X+1)^2 in (Z/7Z)[X] is equal to 2.

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMA><OMS cd="relation1" name="eq"/>
     <OMA><OMS cd="polynomial1" name="coefficient"/>
          <OMA><OMS cd="ring1" name="expression"/>
               <OMA><OMS cd="ring1" name="expression"/>
                    <OMA><OMS cd="ring3" name="poly_ring"/>
                         <OMA><OMS cd="ringname1" name="Zm"/>
                              <OMI>7</OMI>
                         </OMA>
                         <OMV name="X"/>
                    </OMA>
                    <OMA><OMS cd="arith1" name="power"/>
                         <OMA><OMS cd="arith1" name="plus"/>
                              <OMV name="X"/><OMI>1</OMI>
                         </OMA>
                         <OMI>2</OMI>
                    </OMA>
               </OMA>
          </OMA>
     </OMA>
     <OMA><OMS cd="ring1" name="expression"/>
          <OMA><OMS cd="ringname1" name="Zm"/>
               <OMI>7</OMI>
          </OMA>
          <OMI>2</OMI>
     </OMA>
</OMA>
  </OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="polynomial1">coefficient</csymbol>
   <apply><csymbol cd="ring1">expression</csymbol>
    <apply><csymbol cd="ring1">expression</csymbol>
     <apply><csymbol cd="ring3">poly_ring</csymbol>
      <apply><csymbol cd="ringname1">Zm</csymbol><cn>7</cn></apply>
      <ci>X</ci>
     </apply>
     <apply><csymbol cd="arith1">power</csymbol>
      <apply><csymbol cd="arith1">plus</csymbol><ci>X</ci><cn>1</cn></apply>
      <cn>2</cn>
     </apply>
    </apply>
   </apply>
  </apply>
  <apply><csymbol cd="ring1">expression</csymbol>
   <apply><csymbol cd="ringname1">Zm</csymbol><cn>7</cn></apply>
   <cn>2</cn>
  </apply>
 </apply>
</math>

Prefix

eq (coefficient (expression (expression (poly_ring (Zm (7) , X) , power (plus ( X, 1) , 2) ) ) ) , expression (Zm (7) , 2) )

Popcorn

polynomial1.coefficient(ring1.expression(ring1.expression(ring3.poly_ring(ringname1.Zm(7), $X), ($X + 1) ^ 2))) = ring1.expression(ringname1.Zm(7), 2)

Rendered Presentation MathML

$$\mathrm{coefficient}\left(\mathrm{expression}\left(\mathrm{expression}(\mathrm{poly\_ring}(\mathrm{Zm}\left(7\right),X),{(X+1)}^2)\right)\right)=\mathrm{expression}(\mathrm{Zm}\left(7\right),2)$$

Signatures:

sts

expand

Role:

application

Description:

Expands a polynomial.

Example:

The following expression is equivalent to the arithmetic expression X^2+2X+1 in (Z/7Z)[X].

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
    <OMA><OMS cd="polynomial1" name="expand"/>
         <OMA><OMS cd="ring1" name="expression"/>
              <OMA><OMS cd="ring1" name="expression"/>
                   <OMA><OMS cd="ring3" name="poly_ring"/>
                        <OMA><OMS cd="ringname1" name="Zm"/>
                             <OMI>7</OMI>
                        </OMA>
                        <OMV name="X"/>
                   </OMA>
                   <OMA><OMS cd="arith1" name="power"/>
                        <OMA><OMS cd="arith1" name="plus"/>
                             <OMV name="X"/><OMI>1</OMI>
                        </OMA>
                        <OMI>2</OMI>
                   </OMA>
              </OMA>
         </OMA>
    </OMA>
  </OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="polynomial1">expand</csymbol>
  <apply><csymbol cd="ring1">expression</csymbol>
   <apply><csymbol cd="ring1">expression</csymbol>
    <apply><csymbol cd="ring3">poly_ring</csymbol>
     <apply><csymbol cd="ringname1">Zm</csymbol><cn>7</cn></apply>
     <ci>X</ci>
    </apply>
    <apply><csymbol cd="arith1">power</csymbol>
     <apply><csymbol cd="arith1">plus</csymbol><ci>X</ci><cn>1</cn></apply>
     <cn>2</cn>
    </apply>
   </apply>
  </apply>
 </apply>
</math>

Prefix

expand (expression (expression (poly_ring (Zm (7) , X) , power (plus ( X, 1) , 2) ) ) )

Popcorn

polynomial1.expand(ring1.expression(ring1.expression(ring3.poly_ring(ringname1.Zm(7), $X), ($X + 1) ^ 2)))

Rendered Presentation MathML

$$\mathrm{expand}\left(\mathrm{expression}\left(\mathrm{expression}(\mathrm{poly\_ring}(\mathrm{Zm}\left(7\right),X),{(X+1)}^2)\right)\right)$$

Signatures:

sts

degree

Description:

This symbol represents a unary function, whose argument should be univariate polynomial. When applied to a polynomial, it represents its degree, that is the highest power of the variable occurring in a term of the polynomial. If the polynomial has no terms, it is the zero polynomial, in which case the value represented is -1.

Signatures:

sts

leading_term

Description:

This symbol represents a unary function, whose argument should be univariate polynomial. When applied to a polynomial, it represents its leading term, that is the term that is the product of the highest power of the variable and its coefficient. If the polynomial is zero, the value represented is zero.

Signatures:

sts

leading_coefficient

Description:

This symbol represents a unary function, whose argument should be univariate polynomial. When applied to a polynomial, it represents the coefficient of the monomial of highest degree. If the polynomial is zero, the value represented is zero.

Signatures:

sts

leading_monomial

Description:

This symbol represents a unary function, whose argument should be a nonzero univariate polynomial. When applied to such a polynomial, it represents the highest power of the variable occurring in the polynomial.

Commented Mathematical property (CMP):

The leading term of a nonzero polynomial is the product of the leading monomial and the leading coefficient of the polynomial.

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMA><OMS cd="relation1" name="eq"/>
     <OMA><OMS cd="polynomial1" name="leading_term"/>
          <OMV name="f"/>
     </OMA>
     <OMA><OMS cd="arith1" name="times"/>
          <OMA><OMS cd="polynomial1" name="leading_coefficient"/>
               <OMV name="f"/>
          </OMA>
          <OMA><OMS cd="polynomial1" name="leading_monomial"/>
               <OMV name="f"/>
          </OMA>
     </OMA>
</OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="polynomial1">leading_term</csymbol><ci>f</ci></apply>
  <apply><csymbol cd="arith1">times</csymbol>
   <apply><csymbol cd="polynomial1">leading_coefficient</csymbol><ci>f</ci></apply>
   <apply><csymbol cd="polynomial1">leading_monomial</csymbol><ci>f</ci></apply>
  </apply>
 </apply>
</math>

Prefix

eq (leading_term ( f) , times (leading_coefficient ( f) , leading_monomial ( f) ) )

Popcorn

polynomial1.leading_term($f) = polynomial1.leading_coefficient($f) * polynomial1.leading_monomial($f)

Rendered Presentation MathML

$$\mathrm{leading\_term}\left(f\right)=\mathrm{leading\_coefficient}\left(f\right)\mathrm{leading\_monomial}\left(f\right)$$

Signatures:

sts