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How may we write Nth polynomials and perform their derivatives?

Vinícius Hopkins
Posted 2 years ago

Suppose a multivariable polynomial of Nth degree is given by

$P(x,y)=\displaystyle\sum_{j=0}^{N}\sum_{i=0}^{j} a_{i,\,j-i}x^{i}y^{j-i}$.

I want to express this polynomial using Mathematica. In addition, It is of my interest compute its first and second derivatives.

My attempts:

Previously, I tried to study the derivatives of the $P(x,y)$ through a generator of polynomial using the following code:

poly[vars_List, a_, order_] := Module[{n = Length@vars, idx, z}, 
 idx = Cases[Tuples[Range[0, order], n], x_ /; Plus @@ x <= order]; z = Times @@@ (vars^# & /@ idx); z.((Subscript[a, Row[#]]) & /@ idx)]
poly[{x, y}, a, n]

However, it looks like that this code only works when n is equal to some integer number, for example, n =2. Otherwise, I receive an error message.

Attached below, you may find my notebook with my attempts.

Based on the above,

  1. Is there another way to express the general form of $P(x,y)$ as presented above using Mathematica
  2. How may I express the $P(x,y)$ derivatives?

Thanks in advance

Attachments:
Mydoubts.nb
POSTED BY: Vinícius Hopkins
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Rohit Namjoshi
Rohit Namjoshi
Posted 2 years ago

Hi Vinícius

One issue is that N is a built-in symbol. Try

ClearAll@n
poly[{x, y}, a, n]

Range[0, n] is unbounded. Is there a reason why you cannot use Sum to generate the polynomial?
POSTED BY: Rohit Namjoshi

Vinícius,

as Rohit proposed those polynomials and their derivatives can be formulated in a very general way, e.g.:

p = Sum[a[i, j - i] x^i y^(j - i), {j, 0, n}, {i, 0, j}]
(* simple check: *)
p /. n -> 4
(* e.g.: first derivative in x, second in y: *)
dp = D[p, {x, 1}, {y, 2}]
dp /. n -> 4

Does that help? Regards -- Henrik
POSTED BY: Henrik Schachner

You mean you want an expression that is human-readable even when the order is a symbol? Here is an attempt:

poly[vars_List, a_, order_Integer /; order >= 0] :=

 Module[{n = Length@vars, idx, z},
  idx = Select[Tuples[Range[0, order], n], Total[#] <= order &]; 
  Plus @@ 
   Table[Subscript[a, Sequence @@ r] Times @@ (vars^r), {r, idx}]]; 
poly[vars_List /; Length[vars] > 2, a_, order_] :=

 Inactive[Sum][
  Inactive[Times][
   Subscript[a, Subscript["i", 1], \[Ellipsis], 
    Subscript["i", Length[vars]]], vars[[1]]^
   Subscript["i", 1], \[Ellipsis], vars[[-1]]^
   Subscript["i", Length[vars]]], 
  0 <= Inactive[Plus][Subscript["i", 1], \[Ellipsis], 
    Subscript["i", Length[vars]]] <= order]

Try poly[{x, y, z, t}, a, n]
POSTED BY: Gianluca Gorni

Hi Mr. Schachner, I hope you are doing well.

Thank you for your insightful comment. Indeed, I have also tried to compute $P(x,y)$ using Sum. However, it did not work nicely in the general case. Nevertheless, your code has clarified all of my doubts.

Thanks once more, and God bless you.
POSTED BY: Vinícius Hopkins

Thanks for your comments. Your code is quite interesting.

POSTED BY: Vinícius Hopkins
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