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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     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                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                 First derivative of polynomial
    3 12 27 48 75 108

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

       _1 0 0 0 0 1 p. 3
    242

       p. <1 _1 ,. 5 0             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                        Coefficients / Exponents
    _1 2 1
     1 1 1
     2 1 2
     3 0 2

       (<c,.e) p. <y=:2.5 _1       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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