MHB Solving an Unsolved Math Problem: Ring A & Polynomials

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The discussion centers on whether there exists a ring A such that a polynomial of degree 2 can equal a polynomial of degree 4 in A[x]. It is established that, based on the formal definitions of polynomials, a degree 2 polynomial cannot equal a degree 4 polynomial, as their leading coefficients differ. However, it is noted that two polynomials can represent the same polynomial function under certain conditions. An example provided is from the ring of polynomials over the field of integers modulo 2, where p(x) = x^2 and q(x) = x^4 can yield the same value for specific inputs. The conclusion emphasizes the distinction between polynomial equality and functional equivalence.
Fernando Revilla
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I quote an unsolved problem posted in another forum on December 5th, 2012.

Is there any ring A such that in A[x] some polynomial of degree 2 is equal to a polynomial of degree 4? Can you give me an example and explain how this could be true. Thanks in advance!
 
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It is not possible. According to the formal definition of polynomial, if $p$ has degrre $2$ and $q$ degrre 4 then

$$p=(a_0,a_1,a_2,0,\ldots,0,\ldots)\qquad (a_2\neq 0)\\ q=(b_0,b_1,b_2,b_3,b_4,0,\ldots,0,\ldots)\quad (b_4\neq 0)$$

so, $p\neq q$. Another thing is that if $p$ and $q$ can determine the same polynomical function, and the answer is affirmative. For example, choose in $\mathbb{Z}_2[x]$ the polynomials $p(x)=x^2$ and $q(x)=x^4$.
 

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