3 correction of two vectors in the example

A trick which works surprisingly often in my experience: If the Newton polytope of $f$ can not be written as a Minkowski sum of two smaller polytopes, then $f$ is irreducible. I think of this as a generalization of Eisenstein's criterion.

It is surprisingly easy to test whether a lattice polytope in $\mathbb{R}^2$ can be written as a Minkowski sum of smaller lattice polytopes. Let $P$ be a lattice polytope. For example, the convex hull of $(2,0)$, $(1,1)$ and $(0,0)$. Travel around $\partial P$ and write down the vectors pointing from each lattice point to the next. So, in this case, we would write $(-1,1)$, $(-1,-1)$, $(0,1)$, (1,0)$,$(0,1)$. (1,0)$. We'll call this sequence $v(P)$.

It turns out that $v(A + B)$ is simply the sequences $v(A)$ and $v(B)$, interleaved in a certain manner. So, if $P$ can be written as the Minkowski sum $A+B$, we must be able to partition $v(P)$ into two disjoint sub-sequences, each of which sums to zero. In the above example, this can't be done, so any polynomial of the form $a+bx+c x^2 + d xy$, with $a$, $c$ and $d$ nonzero, must be irreducible.

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A trick which works surprisingly often in my experience: If the Newton polytope of $f$ can not be written as a Minkowski sum of two smaller polytopes, then $f$ is irreducible. I think of this as a generalization of Eisenstein's criterion.

It is surprisingly easy to test whether a lattice polytope in $\mathbb{R}^2$ can be written as a Minkowski sum of smaller lattice polytopes. Let $P$ be a lattice polytope. For example, the convex hull of $(2,0)$, $(1,1)$ and $(0,0)$. Travel around $\partial P$ and write down the vectors pointing from each lattice point to the next. So, in this case, we would write $(-1,1)$, $(-1,-1)$, $(0,1)$, $(0,1)$. We'll call this sequence $v(P)$.

It turns out that $v(A + B)$ is simply the sequences $v(A)$ and $v(B)$, interleaved in a certain manner. So, if $P$ can be written as the Minkowski sum $A+B$, we must be able to partition $v(P)$ into two disjoint sub-sequences, each of which sums to zero. In the above example, this can't be done, so any polynomial of the form $a+bx+c x^2 + d xy$, with $a$, $c$ and $d$ nonzero, must be irreducible.

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A trick which works surprisingly often in my experience: If the Newton polytope of $f$ can not be written as a Minkowski sum of two smaller polytopes, then $f$ is irreducible. I think of this as a generalization of Eisenstein's criterion.