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Suppose $F(x_1,\dots,x_n)$ is a homogeneous polynomial in $n$ variables of degree $m$, which has degree $1$ in each of the variables. Suppose further that it has integer relatively prime coefficients. Are there necessary and sufficient conditions on $F$ to guarantee that the equation

$$F(x_1,\dots,x_n) = 1$$

has an integer solution?

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  • $\begingroup$ A special case of this relates to deciding whether a number field is monogenic (in this case $m=n/(n-1)/2$). This was question 6 in Narkiewicz and answered by Uchita, plus given an effective method to check it by Györy. Denis Simons also has a paper for this special case. (This ignores linearity of variables). $\endgroup$
    – pavl0
    Commented May 19, 2019 at 8:10
  • $\begingroup$ @pavl0 can you please clarify? The papers you are referring to all deal with one variable polynomials. $\endgroup$
    – Lenny Fukshansky
    Commented May 20, 2019 at 3:48
  • $\begingroup$ they are dealing homogeneous forms in $n-1$ variables of degree $n(n-1)/2$. $\endgroup$
    – pavl0
    Commented May 20, 2019 at 4:44
  • $\begingroup$ By the way, would linearity imply that $m=n$, as $F(x,\ldots,x)=x^nF(1,\ldots,1)$? $\endgroup$
    – pavl0
    Commented May 20, 2019 at 5:23
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    $\begingroup$ Evidently m\leq n. Wlog F depends on all x_i. Let me assume that for all p, F mod p also depends on all x_i (i.e. for all i, F is not of the form P(x_1, ..., x_{i-1}, k x_i, x_{i+1}, ..., x_n) for P integral). Then write F(x_1, ..., x_n) =: x_n G(x_1, ..., x_{n-1}) + H(x_1, ..., x_{n-1}). Note that G is in n-1 variables and of degree m-1 and satisfies the hypothesis of the question. It suffices to find a solution to G(x_1, ..., x_{n-1}) = 1, since then we may simply solve for x_n. Thus by induction we reduce to the case m = 1, i.e. F is affine linear, in which case it is evident. $\endgroup$
    – alpoge
    Commented May 20, 2019 at 17:45

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