List the set of all formal products of $(1+x^2+x^4)^2(1+x+x^2)^2$ with exponents summing to $4$.
What is this question asking exactly? What is a "formal product"? Does it have anything to do with formal power series? How can be different products of the same expression? What counts as a valid "formal" product of an expression? Obviously if I just simplify the expression I get one thing, not a set. The book seems to not define "formal product", only "product of formal series" which yields a simplified form that I don't think is a set of all possible products.
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$\begingroup$This is not the usual meaning of the term formal product; vadim123 has given that in his answer, but it’s not what appears to be intended here.
If you multiply out $(1+x^2+x^4)^2$ without collecting terms or simplifying products, you get
$$1\cdot1+1\cdot x^2+\underline{1\cdot x^4}+x^2\cdot1+\underline{x^2\cdot x^2}+x^2\cdot x^4+\underline{x^4\cdot1}+x^4\cdot x^2+x^4\cdot x^4\;;$$
the individual terms of this expression are the formal products in question. Three of them, which I’ve underlined, have exponents summing to $4$: $1\cdot x^4$, $x^2\cdot x^2$, and $x^4\cdot 1$. Of course this is only part of the product in your problem, which has many more terms. However, you don’t have to write them all out to pick out the ones with exponents summing to $4$. Just observe that each formal product in
$$(a_1+a_2+\ldots)(b_1+b_2+\ldots)(c_1+c_2+\ldots)\ldots$$
is a product of one term from each of the factors.
$\endgroup$ $\begingroup$Multiplying out the given expression you get a certain polynomial in $x$ of degree $12$. The coefficient of $x^4$ in this polynomial collects all the terms you are interested in, but it is a single number, namely $10$. If you want the individual products leading to terms of exponent $4$ exhibited you can do the following: Define $$a:=(u_01+ u_2 x^2 + u_4x^4)^2(v_01+ v_1 x+ v_2 x^2)^2$$ and determine the coefficient of $x^4$ in the expansion of $a$. Mathematica gives $$u_2^2v_0^2+ 2u_0u_4v_0^2+2u_0 u_2v_1^2+4u_0u_2 v_0v_2+u_0^2v_2^2\ .$$ From this expression you can immediately deduce in which way the $10$ terms of exponent $4$ come about.
$\endgroup$ $\begingroup$The only difference between a formal product and a product is that in the former we are not concerned about whether it represents an actual number; it's just moving symbols and numbers around formally, i.e. by the rules of arithmetic. For example $\sum_{i\ge 0}i!x^i=0!+1!x+2!x^2+3!x^3+\cdots$ is a series that has a radius of convergence 0, but formally we don't worry about which $x$, if any, lead to convergence.
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