File r37/lisp/csl/html/r37_0174.html artifact 3ef0da6ebb part of check-in a57e59ec0d

<A NAME=RESULTANT>

<TITLE>RESULTANT</TITLE></A>
<b><a href=r37_idx.html>INDEX</a></b><p><p>



<B>RESULTANT</B> _ _ _  _ _ _  _ _ _  _ _ _ <B>operator</B><P>
<P>
 
 <P>
<P>
The <em>resultant</em> operator computes the resultant of two polynomials with 
respect to a given variable. If the resultant is 0, the polynomials have 
a root in common. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>resultant</em>(&lt;expression&gt;,&lt;expression&gt;,&lt;kernel&gt;) 
<P>
<P>
<P>
&lt;expression&gt; must be a polynomial containing &lt;kernel&gt; ; 
&lt;kernel&gt; must be a 
<A HREF=r37_0002.html>kernel</A>. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<P><PRE><TT>
resultant(x**2 + 2*x + 1,x+1,x); 

  0 


resultant(x**2 + 2*x + 1,x-3,x); 

  16 


resultant(z**3 + z**2 + 5*z + 5,
          z**4 - 6*z**3 + 16*z**2 - 30*z + 55,
          z);
 

  0 


resultant(x**3*y + 4*x*y + 10,y**2 + 6*y + 4,y); 


   6       5        4        3        2
  Y  + 18*Y  + 120*Y  + 360*Y  + 480*Y  + 288*Y + 64

</TT></PRE><P>The resultant is the determinant of the Sylvester matrix, formed f
rom the 
coefficients of the two polynomials in the following way: 
<P>
<P>
Given two polynomials: 
<P>
<P>
<P><PRE><TT>
    n       n-1 
 a x  + a1 x     + ... + an

</TT></PRE><P>and 
<P>
<P>
<P><PRE><TT>
    m       m-1 
 b x  + b1 x     + ... + bm

</TT></PRE><P>form the (m+n)x(m+n-1) Sylvester matrix by the following means: 
<P>
<P>
<P><PRE><TT>
   0.......0 a  a1 .......... an
   0....0 a  a1 .......... an  0
       .    .   .   .  
   a0 a1 .......... an 0.......0
   0.......0 b  b1 .......... bm
   0....0 b  b1 .......... bm  0
       .    .   .   .  
   b  b1 .......... bm 0.......0  

</TT></PRE><P>If the determinant of this matrix is 0, the two polynomials have a
 common 
root. Finding the resultant of large expressions is time-consuming, due 
to the time needed to find a large determinant. 
<P>
<P>
The sign conventions <em>resultant</em> uses are those given in the article, 
``Computing in Algebraic Extensions,'' by R. Loos, appearing in 
&lt;Computer Algebra--Symbolic and Algebraic Computation&gt;, 2nd ed., 
edited by B. Buchberger, G.E. Collins and R. Loos, and published by 
Springer-Verlag, 1983. 
These are: 
<P>
<P>
<P><PRE><TT>
  resultant(p(x),q(x),x) = (-1)^{deg p(x)*deg q(x)} * resultant(q(x),p(x),x),
  resultant(a,p(x),x)    = a^{deg p(x)},
  resultant(a,b,x)       = 1
</TT></PRE><P>where p(x) and q(x) are polynomials which have x as a variable, an
d 
a and b are free of x. 
<P>
<P>
Error messages are given if <em>resultant</em> is given a non-polynomial 
expression, or a non-kernel variable. 
<P>
<P>
<P>