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nthterm -- the n-th term of a polynomial

Introduction

nthterm(p, n) returns the n-th non-zero term of the polynomial p.

Call(s)

nthterm(p, <vars,> n <, order>)

Parameters

p - a polynomial of type DOM_POLY or a polynomial expression
vars - a list of indeterminates of the polynomial: typically, identifiers or indexed identifiers
n - a positive integer
order - the term ordering: either LexOrder or DegreeOrder or DegInvLexOrder or a user-defined term ordering of type Dom::MonomOrdering. The default is the lexicographical ordering LexOrder.

Returns

a polynomial of the same type as p. An expression is returned if a polynomial expression is used as input. FAIL is returned if n is larger than the actual number of terms of the polynomial.

Overloadable:

p

Related Functions

coeff, degree, degreevec, ground, lcoeff, ldegree, lmonomial, lterm, nterms, nthcoeff, nthmonomial, poly, poly2list, tcoeff

Details

Example 1

We give some self explaining examples:

>> p := poly(100*x^100 + 49*x^49 + 7*x^7, [x]):
   nthterm(p, 1), nthterm(p, 2), nthterm(p, 3)
                    100              49              7
              poly(x   , [x]), poly(x  , [x]), poly(x , [x])
>> nthterm(p, 4)
                                   FAIL
>> nthterm(poly(0, [x]), 1)
                                   FAIL
>> delete p:

Example 2

We demonstrate how the indeterminates influence the result:

>> p := 2*x^2*y + 3*x*y^2 + 6:
   nthterm(p, [x, y], 2), nthterm(p, [y, x], 2)
                                   2   2
                                x y , x  y
>> p := poly(2*x^2*y + 3*x*y^2 + 6, [x, y]):    
   nthterm(p, 2), nthterm(p, [y,x], 2)
                          2                   2
                  poly(x y , [x, y]), poly(y x , [y, x])
>> delete p:

Example 3

We demonstrate the effect of various term orders:

>> p := poly(5*x^4 + 4*x^3*y*z^2 + 3*x^2*y^3*z + 2, [x,y,z]):
   nthterm(p, 1), nthterm(p, 1, DegreeOrder),
   nthterm(p, 1, DegInvLexOrder)
            4                    3    2
      poly(x , [x, y, z]), poly(x  y z , [x, y, z]),
      
               2  3
         poly(x  y  z, [x, y, z])
>> delete p:

Example 4

This example features a user defined term ordering. Here we use the reverse lexicographical order on 3 indeterminates:

>> order := Dom::MonomOrdering(RevLex(3)):
   p := poly(5*x^4 + 4*x^3*y*z^2 + 3*x^2*y^3*z + 2, [x,y,z]):
   nthterm(p, 2, order)
                               3    2
                         poly(x  y z , [x, y, z])

The following call produces all terms:

>> nthterm(p, i, order) $ i = 1..nterms(p)
            2  3                      3    2
      poly(x  y  z, [x, y, z]), poly(x  y z , [x, y, z]),
      
               4
         poly(x , [x, y, z]), poly(1, [x, y, z])
>> delete order, p:

Example 5

The n-th monomial is the product of the n-th coefficient and the n-th term:

>> p := poly(2*x^2*y + 3*x*y^2 + 6, [x, y]):    
   mapcoeffs(nthterm(p, 2), nthcoeff(p, 2)) =
   nthmonomial(p, 2)
                          2                      2
                poly(3 x y , [x, y]) = poly(3 x y , [x, y])
>> delete p:

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