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Let $ \mathbb{F} $ be a field, consider the polynomial ring $ \mathbb{F} \left[ x\right] $ and suppose that the polynomials $f,g \in \mathbb{F} \left[ x\right]$ have degrees $2,2^n$, respectively, where $n$ is a positive integer. Moreover, assume that they are both monic and have exactly one common root.

Question. Is there a known closed form for this common root in terms of the coefficients of $f$ and $g$?

For $n \in \{1,2,3\}$ the root is a rational function with integer coefficients in the coefficients of $f,g$.

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    $\begingroup$ You can perform the Euclidean division of $g$ by $f$; the remainder is a degree 1 polynomial whose zero is your common root. This should give an explicit formula (depending on the degree of $g$). $\endgroup$
    – abx
    Commented Mar 24, 2021 at 13:50
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    $\begingroup$ @abx But if I understand correctly, your proposal is an algorithm, not a closed form expression. $\endgroup$
    – PalmTopTigerMO
    Commented Mar 24, 2021 at 13:54
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    $\begingroup$ You are both right. You need the theory of subresultants which have a closed form like resultants (just the last entry in the subresultant sequence). Essentially these subresultants are the successive remainders of Euclid's algorithm. So if there is exactly one common root you get it from the before last subresultant which is of the form $ax+b$ so your root is $-b/a$. $\endgroup$
    – Abdelmalek Abdesselam
    Commented Mar 24, 2021 at 14:01
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    $\begingroup$ Try §III in Chee Keng Yap, Fundamental problems in algorithmic algebra. $\endgroup$
    – Emil Jeřábek
    Commented Mar 24, 2021 at 14:47
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    $\begingroup$ dl.acm.org/doi/pdf/10.1145/… $\endgroup$
    – Abdelmalek Abdesselam
    Commented Mar 24, 2021 at 16:50

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