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Roots p.  1 1 0 Polynomial

  p. c   ↔ (m;r)
  p.p. c ↔ c

If e is a vector whose elements are all non-negative integers, then p.<c,.e gives the coefficients of the equivalent polynomial:

  (p. <c,.e)&p. ↔ (<c,.e)&p.

 

There are three cases -- coefficients; multiplier and roots; multinomial (boxed matrix of coefficients and exponents):

 c p. x  ↔  +/c*x^i.#c
 (m;r) p. x  ↔  m * */x-r
 (<r)&p.  ↔  (1;r)&p.
 (<c,.e)p.<y  ↔  c+/ .*e*/ .(^~)y

where m is a scalar; c and r are scalars or vectors; and e is a vector or matrix such that ($e)-:(#c),(#y) . A scalar y is extended normally.
 

 

   p. 1 0 0 1
+-+-----------------------------+
|1|_1 0.5j0.866025 0.5j_0.866025|
+-+-----------------------------+

   ]mr=: p. c=: 0 16 _12 2     NB. Multiplier/Roots from Coefficients
+-+-----+
|2|4 2 0|
+-+-----+

   x=: 0 1 2 3 4 5
   (c p. x), ((<c,.i.4)p. x), (mr p. x),: 2*(x-4)*(x-2)*(x-0)
0 6 0 _6 0 30
0 6 0 _6 0 30
0 6 0 _6 0 30
0 6 0 _6 0 30

   c=: 1 3 3 1
   c p. x
1 8 27 64 125 216
   (x+1)^3
1 8 27 64 125 216

   bc=: !~/~i.5                NB. Binomial coefficients
   bc;(bc p./ x);((i.5) ^~/ x+1)
+---------+--------------------+--------------------+
|1 0 0 0 0|1  1  1   1   1    1|1  1  1   1   1    1|
|1 1 0 0 0|1  2  3   4   5    6|1  2  3   4   5    6|
|1 2 1 0 0|1  4  9  16  25   36|1  4  9  16  25   36|
|1 3 3 1 0|1  8 27  64 125  216|1  8 27  64 125  216|
|1 4 6 4 1|1 16 81 256 625 1296|1 16 81 256 625 1296|
+---------+--------------------+--------------------+

   c&p. d. 1 x                 NB. First derivative of polynomial
3 12 27 48 75 108

   (<1 _1 ,. 5 0) p. 3         NB. Coefficients / Exponents
242
              
   _1 0 0 0 0 1 p. 3
242
              
   p. <1 _1 ,. 5 0             NB. Coefficients / Exponents to Coefficients
_1 0 0 0 0 1

   c=: _1 1 2 3 [ e=: 4 2$2 1 1 1 1 2 0 2
   c,.e                        NB. Coefficients / Exponents
_1 2 1
 1 1 1
 2 1 2
 3 0 2
                  
   (<c,.e) p. <y=:2.5 _1       NB. Multinomial
11.75

   c +/ .* e */ .(^~) y
11.75
Note that (<c,.e)p.<y is a “proper” multinomial only if the elements of e are all non-negative integers. In general the powers are not so limited, as in the weighted sum of square root and 4-th root:
   ] t=: <2 3,.1r2 1r4
+-----+
|2 1r2|
|3 1r4|
+-----+

   (t p. 16), +/ 2 3 * 16 ^ 1r2 1r4
14 14
The variant p.!.s is a stope polynomial; it differs from p. in that its definition is based upon the stope ^!.s instead of on ^ (power).


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