Nearly irreducibility of polynomials and the Newton diagrams

Let f be a polynomial in two complex variables. We say that f is nearly irreducible if any two nonconstant polynomial factors of f have a common zero. In the paper we give a criterion of nearly irreducibility for a given polynomial f in terms of its Newton diagram.


Mateusz Masternak
We call a quasi-convenient polynomial f (X, Y ) ∈ C[X, Y ] nondegenerate at infinity if for every real vector w = [p, q] such that p > 0 or q > 0 the system of equations Note that every nearly irreducible polynomial has a connected zero-set. Note that nearly irreducible polynomial may be reducible (e.g. f = XY ). It is easy to check that if f is nearly irreducible and grad f = ( ∂f ∂X , ∂f ∂Y ) = 0 on the curve f (X, Y ) = 0 then f is irreducible (see [17]).
The notion of nearly irreducibility of polynomials in two variables was introduced in [1] by S. Abhyankar and L. A. Rubel in connection with research of these authors on irreducibility of polynomials of the form f (X) − g(Y ). The main result of [1] was reproved by L. A. Rubel, A. Shinzel and H. Tverberg in [17]. Afterwards A. Płoski generalized the result of S. Abhyankar and L. A. Rubel by using the Newton diagram of a given polynomial (see [16], Theorem 2) which is Theorem 1.2 in this note.
is a nonzero vector such that pq 0 then the system of equa- Then the polynomial f is nearly irreducible.
In comparison to Theorem 1.2, in Theorem 1.3 there are no restrictions on the shape of the polygon ∆ ∞ (f ). The proof of Theorem 1.3, based on the Kouchnirenko-Bernstein Theorem, is given in Section 3.
Nearly irreducibility of polynomials and the Newton diagrams

Remark
If there is no pair of parallel faces of the polygon ∆ ∞ (f ) then for any w = 0 at least one of the polynomials in(f, w)(X, Y ) or in(f, − w)(X, Y ) is a monomial and then condition 2 • in our theorem trivially holds, so Theorem 1.3 implies Theorem The examples presented below show that the assumption 2 • in Theorem 1.3 is essential. In particular, Example 2 shows that nonnegative slope of boundary faces of ∆ ∞ (f ) not included in coordinate axes and nondegeneracy at infinity of f are not enough to nearly irreducibility property Example 1 Let us consider the polynomial whose Newton diagram at infinity is drawn in Figure 1.
It is easily seen that the polynomial f is nondegenerate at infinity and that it is not nearly irreducible. Note that condition 2 • of Theorem 1.3 is not satisfied.
whose Newton diagram at infinity is drawn in Figure 2. The polynomial f is nondegenerate at infinity and obviously f is not nearly irreducible. The assumption 2 • of Theorem 1.3 does not hold because if w = [0, 1] then in(f, w)(X, Y ) = Y (X 2 − 1) and in(f, − w)(X, Y ) = X(X 2 − 1) have a common zero in C * × C * . Note that for any c = 0 the polynomial f (X, Y ) + c satisfies 2 • , so it is nearly irreducible.
Its Newton diagram at infinity is given in Figure 3. The polynomial f is nondegenerate at infinity but we can not apply Theorem 1.2 because the polygon ∆ ∞ (f ) possesses faces with positive slope.
is a monomial for any w = 0, hence after Theorem 1.3, the polynomial f is nearly irreducible.

Kouchnirenko-Bernstein Theorem
The famous Bézout theorem for affine curves states that two polynomial equations of given degrees m, n > 0 have at most mn common solutions provided that their number is finite. If additionally their Newton diagrams at infinity are known then we can give more precise estimation. Namely, we may replace the product mn by the Minkowski mixed area of these diagrams. Such results were proved in Kouchnirenko's and Bernstein's papers in 1970s [10,11,12,4]. See also [3,5,8,9]. Focusing only on two-dimensional case much more precise results are possible.
Nearly irreducibility of polynomials and the Newton diagrams then the symbol (f, g) P denotes the intersection multiplicity of f and g at P . We use the definition of the intersection multiplicity as in [7]. We have (f, g) P < +∞ if and only if P is an isolated solution of the given system.
A pair (f, g) of quasi-convenient polynomials is nondegenerate at infinity if for any real vector w = [p, q] such that p > 0 or q > 0 the system of equations [18] for definition) of the diagrams ∆ ∞ (f ) and ∆ ∞ (g), i.e.
where + denotes the Minkowski sum. Let u ∈ ∆ ∞ (f ) and v ∈ ∆ ∞ (g). It is easily seen that there exist points u 1 ∈ ∆(f ), v 1 ∈ ∆(g) and real numbers 0 ≤ s, t ≤ 1 such, that u = su 1 and v = tv 1 . We need to show that u + v = su 1 + tv 1 ∈ ∆ ∞ (f g). The following equality is well-known: ∆(f ) + ∆(g) = ∆(f g) and it holds for every nonzero polynomials f and g (see [14]). Let us note that The quasi-convenience of the polynomials f and g means that their supports have common points with both coordinate axes. Therefore The assumption in Proposition 2.1 about quasi-convenience of the polynomials f and g is essential. [70]

Mateusz Masternak
Let us present a useful version of the Kouchnirenko-Bernstein Theorem in twodimensional case. g) if and only if the pair (f, g) is nondegenerate at infinity.
The first proof of this theorem (in multi-dimensional case) was given by Kouchnirenko in [10] under the additional assumption that the polynomials f and g have identical Newton diagrams at infinity.
The original Bernstein Theorem was formulated for Laurent polynomials in n variables without mentioning Kouchnirenko's assumption and it concerned counting isolated solutions of a system of such polynomials in the set (C * ) n (see [4]). Theorem 2.2 follows from its local version due to Kouchnirenko (i.e. estimation of the intersection multiplicity of plane curves given in terms of their local Newton diagrams, see [10,2,15,6,13]) and from Bézout Theorem for projective curves. For the sake of completeness, we give the proof of Theorem 2.2 in Section 4.

Proof of Theorem 1.3
The proof of our theorem requires two lemmas. Second of them follows from well-known properties of the Minkowski mixed area (see [18], Theorem 5.1.7) but for the convenience of the reader we will give a proof. Proof. Since f (X, 0)f (0, Y ) = 0, we have g(X, 0)g(0, Y )h(X, 0)h(0, Y ) = 0 in C[X, Y ]. Therefore the polynomials g and h are quasi-convenient. Let us suppose, contrary to our claim, that the pair (g, h) is degenerate at infinity. By definition there exists a real vector w = [p, q], where p > 0 or q > 0, such that in(g, w)(x, y) = in(h, w)(x, y) = 0 for some (x, y) ∈ C * × C * . Since g(X, Y ) and h(X, Y ) are coprime divisors of the polynomial f (X, Y ), there exists a polynomial P (X, Y ) such that f (X, Y ) = g(X, Y )h(X, Y )P (X, Y ). Let us note that The above equalities contradict nondegeneracy at infinity of the polynomial f .
if and only if the diagrams ∆ ∞ (f ) and ∆ ∞ (g) form segments included in the same straight line passing through the origin.
In the proof of Lemma 3.2 we need the following Brunn-Minkowski inequality (see [19], Theorem 6.5.3):
It is easy to check that the diagram ∆ ∞ (g) does not contain a point not belonging to the straight line including ∆ ∞ (f ). Indeed, otherwise we would have Area∆ ∞ (f g) > Area∆ ∞ (g). This last observation proves 2 • .
Proof of Theorem 1.3. Let us suppose, contrary to our claim, that there exist poly- Obviously, the polynomials g(X, Y ) and h(X, Y ) are coprime and they are quasiconvenient. From Lemma 3.1 it follows that the pair (g, h) is nondegenerate at infinity. Using now Kouchnirenko-Bernstein Theorem (Theorem 2.2) we state that ν ∞ (g, h) = 0. [72]

Mateusz Masternak
Therefore, Lemma 3.2 implies that the diagrams ∆ ∞ (g) and ∆ ∞ (h) are segments included in the same straight line pα + qβ = 0, where w = [p, q] = 0 and pq 0. So, we have There exists a polynomial P (X, Y ) such that

Proof of Theorem 2.2
Let For every quasi-convenient polynomial we consider additionally its Newton diagram at zero, which is the closure of the set ∆ ∞ (f ) \ ∆(f ) in the natural topology of the real plane. We denote it by ∆ 0 (f ).

Nearly irreducibility of polynomials and the Newton diagrams
The pair (f, g) is nondegenerate at zero if for any real vector w = [p, q] such that p < 0 and q < 0 the system of equations in(f, w)(X, Y ) = in(g, w)(X, Y ) = 0 has no solutions in C * × C * .
Let us recall the estimation of the intersection multiplicity due to Kouchnirenko (see [10,Theorem 1] and [15,Theorem 1.2]). By (f, g) 0 we denote the intersection multiplicity of f and g at O = (0, 0) ∈ C 2 . g) if and only if the pair (f, g) is nondegenerate at zero.
A short and elegant proof of Theorem 4.1 is given in [15], which is based on the Newton-Puisseux theorem. Let us note that the above estimation is interesting only for a pair of quasi-convenient polynomials without constant terms. Indeed, if f (0, 0) = 0 or g(0, 0) = 0, then (f, g) 0 = ν 0 (f, g) = 0 and at the same time the condition of nondegeneracy is trivially satisfied.
Apart from Theorem 4.1, in order to prove Theorem 2.2, we need the next lemma, which follows immediately from Bézout theorem for projective curves.
be coprime polynomials of degree m and n, respectively, and let F (X, Y, Z) and G(X, Y, Z) be their homogenizations. If P 1 = (1:0:0), P 2 = (0:1:0) ∈ P 2 (C), where P 2 (C) is the projective plane, then and (F, G) P2 denote the intersection multiplicity of the projective curves F (X, Y, Z) = 0 and G(X, Y, Z) = 0 at the points P 1 and P 2 , respectively, if and only if the projective curves F (X, Y, Z) = 0 and G(X, Y, Z) = 0 intersect simultaneously the line at infinity L ∞ = {(x:y:z) ∈ P 2 (C) : z = 0} at most at the points P 1 and P 2 .
Nearly irreducibility of polynomials and the Newton diagrams Proof of Theorem 2.2. Let f (X, Y ), g(X, Y ) ∈ C[X, Y ] be quasi-convenient polynomials of positive degrees m and n, respectively.
We may write Hence Since (F, G) P1 = (f 1 , g 1 ) 0 and (F, G) P2 = (f 2 , g 2 ) 0 , using the estimation of intersection multiplicity (Theorem 4.1, 1 • ) we state that (F, G) P1 = (f 1 , g 1 ) 0 ≥ ν 0 (f 1 , g 1 ) and (F, G) P2 = (f 2 , g 2 ) 0 ≥ ν 0 (f 2 , g 2 ). By Bézout Lemma (Lemma 4.2, 1 • ) and the equality (1) we conclude that provided that the polynomials f and g are coprime. So, we proved estimation 1 • of Theorem 2.2. To prove condition 2 • let us note that for any real vector where v = [p − q, −q]. Hence, the pair (f 2 , g 2 ) is nondegenerate at zero if and only if for any real vector w = [p, q] such that q > 0 and q > p, the system of equations in(f, w)(X, Y ) = in(g, w)(X, Y ) = 0 has no solutions in C * × C * . Moreover, let us note that if w = [p, q] and p = q > 0 then the sytem of equations in(f, w)(X, Y ) = in(g, w)(X, Y ) = 0 has no solutions in C * × C * if and only if the projective curves {F (X, Y, Z) = 0} and {G(X, Y, Z) = 0} intersect simultaneously the line at infinity L ∞ at most at the points P 1 and P 2 .
Note that if the pair (f, g) is nondegenerate at infinity then f and g are coprime. Since (F, G) P1 = (f 1 , g 1 ) 0 and (F, G) P2 = (f 2 , g 2 ) 0 , to finish the proof of 2 • of Theorem 2.2, it is enough to apply 2 • of Theorem 4.1 to the pairs (f 1 , g 1 ) and (f 2 , g 2 ) and to use Bézout Lemma (Lemma 4.2, 2 • ) and the equality (1).