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I have tried to start with a polynomial with $2$ variables and with degree $2$; it was simple.

But with degree $n$ is much harder. I would like to know the general form not only with $2$ variables but with $m$ variables.

hardmath
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Lamar A. Chidiac
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  • I retagged your post to remove the tags "abstract-algebra" and "number-theory", because these tags are not related to what you're asking. – Toby Mak Oct 21 '17 at 10:44
  • Here's a hint - the general form of a polynomial is $a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$, where $a_n$ stands for the coefficient with power $n$. How many coefficients must you set equal to $0$ so that the polynomial has $n$ non-zero variables? – Toby Mak Oct 21 '17 at 10:45
  • @TobyMak I didn't get it honestly. I think you understand me wrong, i want a polynomial with variables x1,x2,....xm and each variable can be to the power of i<=n – Lamar A. Chidiac Oct 21 '17 at 10:54
  • Can you rewrite what you have said? I don't understand you either. – Toby Mak Oct 21 '17 at 10:55
  • @TobyMak The question is not asking for polynomials with $m$ terms. No. It is asking for polynomials with $m$ variables. – Zach Teitler Oct 21 '17 at 11:12
  • Well, if the question is as you said, then how can there be a general form when you can choose so many combinations of letters to use? – Toby Mak Oct 21 '17 at 11:14
  • @LamarA.Chidiac If you are asking how many terms there will be, then take a look at https://math.stackexchange.com/questions/36250/number-of-monomials-of-certain-degree – Zach Teitler Oct 21 '17 at 11:19

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Hint: $$\sum_{i_1+i_2+i_3+…i_m\le n}a_{i_1i_2i_3…i_m}x_1^{i_1}x_2^{i_2}…x_m^{i_m}$$ as ${i_k} $ varies through non-negative integers...

Macavity
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