Proof of Proposition 3.2.

Write $I=I_{1}\cup I_2$ where $I_1=\{L\in I\mid N_{\mathrm {aff}}(L,B)\leq 1 \}$ and $I_2=\{L\in I\mid N_{\mathrm {aff}}(L,B)>1\}$ . It is clear that $N_{\mathrm {aff}}(X\cap \cup _{L\in I_1}L)\leq \#I_1$ . If $L\in I_2$ , then there exist $a=(a_1,a_2,a_3), v=(v_1,v_2,v_3)\in {\mathbb {Z}}^{3}$ such that $H(a):=\max _i |a_i|\leq B$ , v is primitive and $L({\mathbb {Q}})=\{a+t v\mid t\in {\mathbb {Q}}\}$ . Since v is primitive we deduce that

$$ \begin{align*}L({\mathbb{Z}})\cap [-B,B]^{3}=\{a+t v\mid t\in {\mathbb{Z}},\ H(a+t v)\leq B\} \end{align*} $$

and thus since $L\in I_2$ it follows that $H(v)\leq 2B$ , so we have

$$\begin{align*}N_{\mathrm{aff}}(L ,B)=\#(L({\mathbb{Z}})\cap [-B,B]^{3}) \leq 1+\dfrac{2B}{H(v)} \le \dfrac{4B}{H(v)}. \end{align*}$$

We also have $f_{|L} = 0$ , therefore when we write $f(a+tv)=\sum \limits _{i=0}^d c_i(a,v) t^i$ , it follows that $c_i(a,v)=0$ for every $0\le i \le d$ . Explicitly,

$$\begin{align*}f(a+t v) = t^d f_d(v) + t^{d-1}\underbrace{\left(\sum a_i \frac{df_d}{dx_i}(v)+f_{d-1}(v)\right)}_{:=c_{d-1}(a,v)} +\ldots + t \underbrace{\sum v_i \frac{df}{dx_i}(a)}_{:=c_{1}(a,v)}+ f(a)\equiv 0. \end{align*}$$

Let C denote the curve in $\mathbb {P}^2_{{\mathbb {Q}}}$ cut out by $f_d$ and fix $v_0 \in C({\mathbb {Q}})$ . We count the number of lines in I in the direction of $v_0$ . Because of the NCC condition, $c_1(a,v_0)$ is a nonzero polynomial in a. Indeed, otherwise there exists a linear change of variables such that f only depends on $2$ variables. Hence, there are at most $d(d-1)$ lines on X in the direction of $v_0$ , since they must all lie on the intersection of X with the surface cut out by $c_1(a,v_0) = 0$ , which is a $1$ -dimensional variety of degree $d(d-1)$ . Factor $f_d = \prod _\ell g_\ell $ over ${\mathbb {Q}}$ , where $g_\ell $ is irreducible of degree $d_\ell $ and set

$$\begin{align*}A_{\ell, i} = \{v\in {\mathbb{P}}^{2}_{\mathbb{Q}}({\mathbb{Q}})\mid g_\ell(v) = 0,\ \text{ and } H(v) = i\}, \end{align*}$$

and $n_{\ell ,i} = \# A_{\ell ,i}$ . Then by Theorem 2.3, we have that

(3.3) $$ \begin{align} \sum_{1\leq i\leq k}n_{\ell,i}\ll d_{\ell}^{2}k^{\frac{2}{d_\ell}}(\log k)^{O(1)}. \end{align} $$

At the same time, for each $i\ge 1$ the Schwartz–Zippel bound gives us

(3.4) $$ \begin{align} n_{\ell,i} \ll d_\ell i. \end{align} $$

By our discussion,

$$ \begin{align*}N_{\mathrm{aff}}(X \cap ( \cup_{L \in I} L ) ,B) \leq (\#I_1) +d(d-1)\sum_\ell \sum_{i=1}^{2B}n_{\ell,i}\dfrac{4B}{i}. \end{align*} $$

It is now left to bound $\sum _{i=1}^{2B} n_{\ell ,i}\frac {B}{i}$ , for each $\ell $ . Let us first assume that $f_d$ is $2$ -irreducible, so that $d_\ell \geq 3$ for every $\ell $ . From Inequality (3.4) we find,

$$ \begin{align*}\sum_{i=1}^{d_\ell}n_{\ell,i}\dfrac{4B}{i} \ll \sum_{i=1}^{d_\ell}d_\ell i\dfrac{4B}{i} \ll \sum_{i=1}^{d_\ell} d_\ell B \le d_\ell^2 B. \end{align*} $$

On the other hand, summation by parts yields:

(3.5) $$ \begin{align} \sum_{i=d_\ell}^{2B}n_{\ell,i} \dfrac{4B}{i}&=\sum_{k=d_\ell}^{2B-1}\left(\sum_{i=d_\ell}^{k}n_{\ell,i} \right) \left(\dfrac{4B}{k}-\dfrac{4B}{k+1}\right)+\left(\sum_{i=d_\ell}^{2B}n_{\ell,i} \right)\dfrac{4B}{2B} \end{align} $$

(3.6) $$ \begin{align}\kern90pt &\ll d_\ell^2\left( \left( \sum_{k=d_\ell}^{2B-1}k^{\frac{2}{d_\ell}} \log(k)^{O(1)} \dfrac{B}{k(k+1)} \right ) +B^{\frac{2}{d_\ell}} \log(B)^{O(1)} \right). \end{align} $$

Now since $d_\ell \geq 3$ , one has (note that $d_\ell ^{1/d_\ell }$ is universally bounded):

(3.7) $$ \begin{align} \sum_{k\geq d_\ell} \frac{ k^{\frac{2}{d_\ell}} \log(k)^{O(1)} }{k(k+1)}\ll \sum_{k\geq d_\ell} \frac{ \log(k)^{O(1)}}{{k^{2-2/d_\ell}} } \ll \frac{d_\ell^{2/d_\ell} \log(d_\ell)^{O(1)}} { d_\ell } \ll d_\ell^{-1} \log(d_\ell)^{O(1)}, \end{align} $$

and hence one finds

$$ \begin{align*}\sum_{i=d_\ell}^{2B}n_{\ell,i}\dfrac{4B}{i} \ll d_\ell\log(d_\ell)^{O(1)}B + d_\ell^2B^{2/d_\ell}\log(B)^{O(1)} \ll d_\ell^2B. \end{align*} $$

Thus we conclude the correct upper bound in this case, proving the proposition when $f_d$ is $2$ -irreducible. When $f_d$ is only $1$ -irreducible (instead of $2$ -irreducible), one estimates (3.6) by using

$$ \begin{align*}\sum_{k = 1 }^{2B-1} \frac{\log(k)^{O(1)}}{k+1}\ll \log (B)^{O(1)} \end{align*} $$

instead of (3.7). The proposition is now fully proved.

Proof of Proposition 3.1.

We start by using Proposition 2.1 on f to find an auxiliary polynomial g of degree at most $O(1) d^{5/2} B^{1/\sqrt {d}}(d+\log (B))$ which is not divisible by f, and which vanishes on all integral points of $V(f)$ of height at most B. We now consider two cases, depending on whether d or B is large.

Assume first that $d> \log (B)^2$ . Then $B^{1/\sqrt {d}} = O(1)$ is universally bounded, hence $g = f = 0$ is a (possibly reducible) curve of degree at most

$$\begin{align*}\ll d^{9/2}. \end{align*}$$

The Schwartz–Zippel bound then gives that

$$\begin{align*}N_{\mathrm{aff}}(f,B) \leq N_{\mathrm{aff}}(C, B)\ll d^{9/2} B. \end{align*}$$

Now assume that $d \leq \log (B)^2$ . Hence, $g=f=0$ is a (possibly reducible) curve C of degree at most

$$\begin{align*}\deg(C) \ll d\cdot d^{5/2} B^{1/\sqrt{d}}(d+\log(B))= B^{1/\sqrt{d}}\log(B)^{O(1)}. \end{align*}$$

By Proposition 3.2, the contribution of the union of all linear irreducible components of C is at most $O(1)d^{4}B$ if $f_d$ is $2$ -irreducible, and at most $O(1)d^4 B\log (B)^{O(1)}$ if $f_d$ is $1$ -irreducible (instead of $2$ -irreducible). By Corollary 2.2, the total contribution of components $C_i$ of degree $ 2 \leq \deg (C_i) \leq \log B$ , is bounded by

$$ \begin{align*} \ll \deg(C)(\log B)^3 B^{1/2}(\log (B)+\log B) &\ll B^{1/\sqrt{d}}B^{1/2}(\log B)^{O(1)}. \end{align*} $$

Distinguishing between $d\geq 5$ , $d = 4$ , and $d=3$ , this quantity is bounded by

$$ \begin{align*} &\ll B, & \text{ if } d \geq 5, \\ &\ll B \log(B)^{O(1)}, & \text{ if } d =4, \\ &\ll B^{1/2 + 1/\sqrt{3}} \log(B)^{O(1)}, & \text{ if } d =3. \end{align*} $$

Finally, for each irreducible component $C_i$ of C with $d_i:=\deg (C_i)> \log (B)$ , we have by Theorem 2.3 that

$$\begin{align*}N_{\mathrm{aff}}(C_i, B) \ll d_i^2 B^{1/d_i}(\log B)^{O(1)} \ll d_i^2 B^{1/\log(B)} \log(B)^{O(1)} \ll d_i^2 \log(B)^{O(1)}. \end{align*}$$

Summing over all irreducible components $C_i$ of C with $d_i:=\deg (C_i)> \log (B)$ ,

$$ \begin{align*} \sum_i N_{\mathrm{aff}}(C_i, B) &\ll \sum_i d_i^2 \log(B)^{O(1)} \le \log(B)^{O(1)}\Big(\sum_i d_i\Big)^2 \\ &\le \log(B)^{O(1)}\deg(C)^2 \ll B^{2/\sqrt{d}} \log(B)^{O(1)}. \end{align*} $$

Again, this is $O(B)$ for $d\geq 5$ , while it is also bounded as desired for $d=3$ and $d=4$ . This concludes the proof.

Proof. Any such plane must be contained in the tangent cone $T_x$ of X at x. If X is not a cone with cone point x, then the intersection $T_x\cap X$ is of dimension $n-2$ and degree at most $d^2$ . If x is a smooth point, then the tangent cone is simply the tangent space, and the intersection $T_x\cap X$ is of degree d.

Proof of Lemma 3.6.

The singular locus $X_{\mathrm {sing}}$ is contained in a subvariety of ${\mathbb {P}}^n$ of (pure) dimension $n-2$ and degree $d^2$ , and hence contains at most $d^2$ linear spaces of dimension $(n-2)$ . Let Z be the union of all isolated $(n-2)$ -planes on X which are not contained in $X_{\mathrm {sing}}$ , and denote by Y the closed set $Z\cap X_{\mathrm {sing}}$ . This is an algebraic set of dimension at most $n-3$ . We now take a generic $2$ -plane $P\subset {\mathbb {P}}^n$ such that $P\cap Y = \emptyset $ , such that $P\cap X$ is geometrically integral, such that P is not contained in the hyperplane $\{x_0 = 0\}$ , and such that the intersection of $\{x_0 = 0\}$ and P does not contain any singular points of X. If L is an isolated $(n-2)$ -plane on X which is not contained in $X_{\mathrm {sing}}$ , then by construction L intersects P in a unique point in $X_{\mathrm {ns}}$ . So it suffices to count the number of $x\in X_{\mathrm {ns}}\cap P$ for which there exists an isolated $(n-2)$ -plane on X through x. Indeed, Lemma 3.5 shows that the total number of isolated $(n-2)$ -planes contained in X is at most d times this quantity plus $d^2$ (coming from $(n-2)$ -planes in $X_{\mathrm {sing}}$ ).

Now take $x\in X_{\mathrm {ns}}\cap P$ and assume that there is some $(n-2)$ -plane L on X through x. Then L must also be contained in the tangent space $T_x X$ . Let $f(x_0, \ldots , x_n)\in \overline {{\mathbb {Q}}}[x_0, \ldots , x_n]$ be a defining equation for X, and assume without loss of generality that $\frac {\partial f}{\partial x_0} (x) \neq 0$ . Define the polynomial

$$\begin{align*}g(x,a) := f\left(-\frac{\partial f}{\partial x_1}(x) a_1 - \ldots -\frac{\partial f}{\partial x_n}(x) a_n, \frac{\partial f}{\partial x_0}(x) a_1, \ldots, \frac{\partial f}{\partial x_0}(x) a_n\right) \end{align*}$$

in $\overline {{\mathbb {Q}}}[x_0, \ldots , x_n][a_1, \ldots , a_n]$ . Then X contains an $(n-2)$ -plane through x if and only if the polynomial $g(x,a)\in \overline {{\mathbb {Q}}}[a_1, \ldots , a_n]$ has a linear factor in $\overline {{\mathbb {Q}}}[a_1, \ldots , a_n]$ . Let K be the function field of the geometrically integral curve $C=P\cap X_{\mathrm {ns}}$ (over $\overline {{\mathbb {Q}}}$ ). By multiplying with some negative power of $x_0$ , we can consider g as an element of $\overline {K}[a_1, \ldots , a_n]$ . Each coefficient of g is of degree at most $d(d-1)$ in $\overline {{\mathbb {Q}}}[x]$ , and hence has height at most $d^2(d-1)$ in K. We factor g in this ring into irreducible polynomials $g_i\in \overline {K}[a_1, \ldots , a_n]$ . The number of $g_i$ which are linear is equal to the number of one-dimensional irreducible components of $F_{n-2}(X)$ . Since we are only considering isolated $(n-2)$ -planes, we can disregard these and assume that $\deg g_i = \delta _i \geq 2$ for all i. We fix i for the moment, and let $\{F_\ell \}_\ell $ be a collection of Noether forms for polynomials of degree $\delta _i$ from Theorem 2.5, for which we have that $\deg F_\ell \leq \delta _i^2$ . Since $g_i$ is irreducible over $\overline {K}$ , not all $F_\ell $ vanish when applied to $g_i$ . So let $F_\ell $ be nonvanishing on $g_i$ and denote by $\tilde {g}\in K$ the product of all conjugates of $F_\ell (g_i)$ . We bound the height of this $\tilde {g}$ , which will give us an upper bound on its number of zeroes. By the discussion above, we have that $\sum _i h(g_i) \leq h(g)$ and hence $h(g_i)\leq h(g)$ . Now note that all poles of all coefficients of g are contained in the intersection of C with the line defined by $x_0 = 0$ in P, which consists of at most d points. By Gauss’ lemma, all poles of all coefficients of the $g_i$ therefore lie above these points, and so we obtain that $h(F_\ell (g_i))\leq d \delta _i^2 d^2(d-1)$ . Hence $h(\tilde {g})\leq \delta _i d^5$ . Now sum over all i to conclude that there are at most $d^6$ points x in C for which g has a linear factor. For each such point, there can be at most d isolated $(n-2)$ -planes on X containing this point, and hence we conclude that X contains at most $d^7 + d^2$ isolated $(n-2)$ -planes.

Proof. If X is $(n-2)$ -ruled, then so is its projective closure $\overline {X}$ in ${\mathbb {P}}^n$ as it contains infinitely many $(n-2)$ -planes. It follows $X_\infty $ is $(n-3)$ -ruled. Let Y be an irreducible component of $X_\infty $ and take $y \in Y$ which does not lie on any other irreducible components of $X_\infty $ . There exists an $(n-3)$ -plane contained in $X_\infty $ and passing through y, which must then lie on Y by irreducibility. The claim now follows as Y must contain infinitely many $(n-3)$ -planes.

Proof of Proposition 3.4.

Recall that $X=V(f) \subset {\mathbb {A}}^n_{{\mathbb {Q}}}$ . By Lemma 3.6, the number of isolated $(n-2)$ -planes on X is at most $O(d^7)$ . These have at most $O_n(d^7B^{n-2})$ integral points up to height B, and so we can focus on the one-dimensional components of $F_{n-2}(X)$ . In particular, we can assume that X is $(n-2)$ -ruled.

Let Y be an irreducible component of $X_\infty $ of degree $\delta $ , which is $(n-3)$ -ruled by the previous lemma. So the Fano scheme $F_{n-3}(Y)$ of $(n-3)$ -planes on Y is of dimension one. Let $C\subset F_{n-3}(Y)$ be an irreducible component of $F_{n-3}(Y)$ of dimension one which $(n-3)$ -rules Y. We claim that

$$\begin{align*}\delta \leq \deg C. \end{align*}$$

To prove this, let $\ell $ be a generic line in ${\mathbb {P}}^{n-1}$ for which $\ell \cap Y$ consists of $\delta $ points. Then

$$\begin{align*}H := \{L\in \mathrm{Gr}(n-2, n)\mid L\cap \ell \neq \emptyset \} \end{align*}$$

is a hyperplane in $\mathrm {Gr}(n-2,n)$ . We claim that C is not contained in H. Suppose towards a contradiction that $C\subset H$ . Then the regular map

$$\begin{align*}C\to \ell: L\mapsto L\cap \ell \end{align*}$$

is well-defined, and the image is the finite set $\ell \cap Y$ as C rules Y. But C is irreducible, and so the image is a single point, contradicting that $\ell \cap Y$ consists of $\delta \geq 2$ points. So C is not contained in H and $\#H\cap C\leq \deg C$ . Therefore

$$\begin{align*}\delta = \#\ell\cap Y\leq \#H \cap C \leq \deg C, \end{align*}$$

proving the lower bound.

Conversely, we claim that the sum of the degrees of the one-dimensional components of $F_{n-3}(Y)$ is at most $\delta ^2$ . Denote this union of all one-dimensional components of $F_{n-3}(Y)$ by D. To prove the claim, we take a generic hyperplane H in $\mathrm {Gr}(n-2,n)$ , which corresponds to a line $\ell $ in ${\mathbb {P}}^{n-1}$ as above. Then $\ell $ is also generic (it is not contained in Y as $H \cap C$ is finite, and Y is ruled), and so we can assume that $\ell $ intersects Y in $\delta $ smooth points. If $L\in H\cap C$ then we obtain an $(n-3)$ -plane on Y through a point on $\ell \cap Y$ . By Lemma 3.5 there are at most $\delta ^2$ such planes, so that

$$\begin{align*}\#H\cap D = \deg D\leq \delta^2. \end{align*}$$

We have now proven the claim.

If $L\subset X$ is an $(n-2)$ -plane, then (see, for example, [Reference BrowningBro09, Lemma 4.5])

$$\begin{align*}N_{\mathrm{aff}}(L, B)\ll_n \frac{B^{n-2}}{H(L)} + B^{n-3}, \end{align*}$$

where $H(L)$ is the height of the corresponding point in projective space coming from the Plücker embedding of $L\cap {\mathbb {P}}^{n-1}\in \mathrm {Gr}(n-2, n)$ . If $H(L)> B$ then $N_{\mathrm {aff}}(L,B)\ll _n B^{n-3}$ , and so it suffices to treat those L with $H(L)\leq B$ . Each of these gives a rational point on $F_{n-3}(Y)$ of height at most B, for some component Y of $X_\infty $ . So using Lemma 3.5 the count is

$$ \begin{align*} N_{\mathrm{aff}}(X \cap ( \cup_{L \in I} L ) ,B) &\ll_n \#I B^{n-3} + d^2 \sum_{C} \sum_{\substack{L\in C({\mathbb{Q}})\\ H(L)\leq B}} \frac{B^{n-2}}{H(L)}, \end{align*} $$

where the sum is over all C which are one-dimensional components of $F_{n-3}(Y)$ for some component Y of $X_\infty $ . Now one proceeds exactly as in the proof of Proposition 3.2 to conclude.

Proof of Theorem 1.5.

Recall that we are given $f\in {\mathbb {Z}}[x_1, \ldots , x_n]$ of degree $d\geq 3$ , and we wish to bound the number of integral points x of height at most B satisfying the equation $f(x)=0$ . We use induction on n, where the base case $n=3$ is given by Proposition 3.1. We therefore assume that $n \ge 4$ , and let $\kappa = 5(n-2) = O_n(1)$ , where $5n-12=5(n-2)-2$ is the constant from Proposition 2.7 when $m=n-2$ .

First assume that $d>\kappa ^{n-1}$ . Let g be the auxiliary polynomial from Proposition 2.1 applied to f, it has degree at most

$$\begin{align*}\ll_n d^{4-\frac{1}{n-1}} B^{\frac{1}{d^{1/(n-1)}}} \log B, \end{align*}$$

is coprime to f and vanishes on all integral points of $f=0$ of height at most B. Let C be the variety cut by $f=g=0$ , it is of dimension $n-2$ and has degree at most $O_n(1) d^{5-\frac {1}{n-1}} B^{\frac {1}{d^{1/(n-1)}}} \log B$ . Let $C_i$ be the irreducible components of C, and denote by $d_i$ the degree of $C_i$ .

Assume that $d < (\log B)^{n-1}$ . By Proposition 3.4 the contribution to $N_{\mathrm {aff}}(f,B)$ coming from the components $C_i$ with $d_i = 1$ is at most $O_n(1) d^7 B^{n-2}$ if $f_d$ is $2$ -irreducible and at most $O_n(1) d^7 B^{n-2} \log (B)$ if $f_d$ is $1$ -irreducible (instead of $2$ -irreducible). Using Proposition 2.7, the contribution to $N_{\mathrm {aff}}(f, B)$ coming from components $C_i$ with $2\leq \delta _i\leq \log B$ is at most

$$\begin{align*}O_n(1) d^{5-\frac{1}{n-1}}(\log B)^\kappa B^{n-3 + \frac{1}{2} + \frac{1}{d^{1/(n-1)}}}. \end{align*}$$

By our assumption that $d> \kappa ^{n-1}\geq 3^{n-1}$ this is indeed bounded by $O_n(1) B^{n-2}$ . On the other hand, if $\delta _i> \log B$ then $B^{\frac {1}{\log B}} = O(1)$ and so using Proposition 2.7, the contribution from components $C_i$ with $\delta _i> \log B$ is at most

$$\begin{align*}O_n(1) B^{n-3} (d^{5-\frac{1}{n-1}})^\kappa B^{\frac{\kappa}{d^{1/(n-1)}}}. \end{align*}$$

By our assumption that $d> \kappa ^{n-1}$ , this quantity is also bounded by $O_n(1) B^{n-2}$ .

Now assume that $d> (\log B)^{n-1}$ , but still that $d> \kappa ^{n-1}$ . Then the Schwartz–Zippel bound immediately gives that

$$\begin{align*}N_{\mathrm{aff}}(C, B) \ll_n d^{5-\frac{1}{n-1}} B^{\frac{1}{d^{1/(n-1)}}}B^{n-2} \ll_n d^{5 - \frac{1}{n-1}} B^{n-2} \end{align*}$$

since $B^{1/d^{1/(n-1)}}$ is bounded by a universal constant.

Finally, assume that $d \leq \kappa ^{n-1} = O_n(1)$ . By Corollary 3.8, we may find linearly independent ${\mathbb {Q}}$ -linear forms $\ell ,\ell '$ such that after a change of coordinates $x_n:=\ell (x)$ , $x_{n-1}:=\ell '(x)$ of height $O_n(d^3) = O_n(1)$ , there exists an integer t of height at most $O_n(d^3)=O_n(1)$ such that the polynomial $f_d(x_1, \ldots , x_{n-1}, tx_{n-1})$ is $2$ -irreducible (or $1$ -irreducible if $f_d$ was) and such that there are at most $O_n(d) = O_n(1)$ values $b\in {\mathbb {Q}}$ for which $f(x_1, \ldots , x_{n-1}, tx_{n-1} + b)$ is not NCC. Now we use the induction hypothesis for the good values of b and the Schwartz–Zippel bound for the bad values of b to get

$$ \begin{align*} N_{\mathrm{aff}}(f, B)&\leq \sum_{|b|\leq (|t|+1)B} N_{\mathrm{aff}}\left( \{f=0\}\cap \{x_n = tx_{n-1} + b\}, B\right) \\ &\ll_n d^3B(d^5B^{n-3}) + d^2B^{n-2} \ll_n B^{n-2}. \end{align*} $$

If $f_d$ is $1$ -irreducible, the induction hypothesis introduces an additional factor of the form $\log (B)^{O_n(1)}$ in the upper bound above.

Proof of Theorem 1.9.

Let $f\in {\mathbb {Z}}[x_1, \ldots , x_n]$ be of degree d such that $f_d$ is $1$ -irreducible. We may assume that f is irreducible. We induct on n, where the base case $n=2$ is Conjecture 1.8. So assume that $n\geq 3$ . First, if $\lVert f\rVert $ is larger than $O_n\left (B^{d\binom {d+n-1}{n-1}})\right )$ , then we can use [Reference Browning, Heath-Brown and SalbergerBHBS06, Lem. 5] to bound $N_{\mathrm {aff}}(f, B)$ as desired. So we may assume that $\log \lVert f\rVert \ll _n d^{O_n(1)}\log B$ . By Corollary 1.11 with $r=m=s=1$ , we can find $t\in {\mathbb {Z}}$ with $|t| \leq \mathrm {poly}_{n,d}(\log \lVert f\rVert ) \le \mathrm {poly}_{n,d}(\log B)$ such that $f_d(x_1, \ldots , x_{n-1}, tx_{n-1})$ is still $1$ -irreducible. By induction, we have

$$ \begin{align*} N_{\mathrm{aff}}(f, B)&\leq \sum_{|b|\leq (|t|+1)B} N_{\mathrm{aff}}\left( \{f=0\}\cap \{x_n = tx_{n-1} + b\}, B\right) \ll_{n,d,\varepsilon} B^{n-2+\varepsilon}.\\[-51pt] \end{align*} $$

Proof of Theorem 1.2.

Consider the affine cone $X_{\mathrm {aff}}$ over X, which is an affine hypersurface inside ${\mathbb {A}}^{n+1}_{{\mathbb {Q}}}$ (given by the same equation as X), and use Theorem 1.5 for $X_{\mathrm {aff}}$ . We are done since clearly $ N(X,B)\le N_{\mathrm {aff}}(X_{\mathrm {aff}},B). $

Proof. We first prove the case $r=1$ . Set $T_1=T$ , put $a = 1+\max _i \deg _{Y_i} F$ and consider the Kronecker transform of F

$$\begin{align*}\operatorname*{\mathrm{{Kr}}}(F) = F(T, Y, Y^a, \ldots, Y^{a^{n-1}})\in {\mathcal{O}}_K[T,Y]. \end{align*}$$

Note that $H_K(\operatorname *{\mathrm {{Kr}}}(F)) = H_K(F)$ , that $\deg _T(\operatorname *{\mathrm {{Kr}}}(F)) = \deg _T(F)$ , and that

$$\begin{align*}\deg_{\underline{Y}} (\operatorname*{\mathrm{{Kr}}}(F)) \leq (a-1) + (a-1)a + \ldots (a-1)a^{n-1} = a^n - 1 < (d+1)^n. \end{align*}$$

Let

$$\begin{align*}\operatorname*{\mathrm{{Kr}}}(F) = \prod_{i=1}^m Q_i(T,Y) \end{align*}$$

be an irreducible factorization of $\operatorname *{\mathrm {{Kr}}}(F)$ in $K[T,Y]$ . We have that $m\leq \deg _{Y} (\operatorname *{\mathrm {{Kr}}} (F))$ and for each i that

$$\begin{align*}\deg_T (Q_i) \leq \deg_T (\operatorname*{\mathrm{{Kr}}} (F)), \quad \deg_{Y} (Q_i) \leq \deg_{Y} (\operatorname*{\mathrm{{Kr}}} (F)). \end{align*}$$

Also, by combining [Reference LangLan83, Ch.3, Prop. 2.4] and [Reference Paredes and SasykPS24, Lemma 2.1] we have

$$\begin{align*}H_K(Q_i) \ll_{K,n} H_K(Q_i) H_K(\prod_{j\not=i} Q_j) \ll_{K,n} 4^{d_K(\deg (\operatorname*{\mathrm{{Kr}}}(F))+1)^2} H_K(\operatorname*{\mathrm{{Kr}}}(F)). \end{align*}$$

We now follow the proof of [Reference Fried and JardenFJ23, Lem. 13.1.3] (see the note right after [Reference Fried and JardenFJ23, Equation (13.2)]), which gives for each nonempty proper subset $I\subset \{1, \ldots , m\}$ a nonzero polynomial $c_I(T)\in K[T]$ such that $\deg c_I \leq \deg _T F$ with the following property: let $t_0\in K$ and assume that for each $i = 1, \ldots , m$ , the polynomial $Q_i(t_0, Y)$ is irreducible in $K[Y]$ , and for each nonempty proper subset $I\subset \{1, \ldots , m\}$ , we have $c_I(t_0)\neq 0$ , then $F(t_0, Y_1, \ldots , Y_n)$ is irreducible in $K[Y_1, \ldots , Y_n]$ . In other words, we have that

(*) $$\begin{align} S_T(F, B) \leq \sum_{i=1}^m S_T(Q_i, B) + \sum_{I \subset \{1,\ldots,m \}} \deg c_I, \end{align}$$

where $\deg _{Y} Q_i \le \deg _{Y} \operatorname *{\mathrm {{Kr}}}(F) \le (d+1)^n$ and $\deg _T Q_i,\deg c_I \le d$ . We conclude using [Reference Paredes and SasykPS24, Thm. 1.1] to bound $S_T(Q_i, B)$ , recalling $m \le (d+1)^n$ :

$$\begin{align*}(*) \ll_{K,n} (d+1)^n2^{36(d+1)^n}(d+1)^{35n} d^{26}(\log H_K(F)+1)^{10} (d+1)^{20n}B^{1/2} +2^{(d+1)^n}d. \end{align*}$$

To prove the multivariate version we proceed by induction, where the base case is $r=1$ . Assume that $r>1$ , and consider specializations of $T_r$ . Using the $r=1$ case we get,

$$ \begin{align*} S_{T}(F,B) \ll_K S_{T_r}(F,B)B^{r-1} + \sum_{t_r \in [B]_{{\mathcal{O}}_K}} S_{(T_1,\ldots,T_{r-1})}(F|_{T_r=t_r},B). \end{align*} $$

Note that for every $t_r \in [B]_{{\mathcal {O}}_K}$ we may bound the height of the specialization $F|_{T_r=t_r}$ ,

$$\begin{align*}H_K(F|_{T_r=t_r}) \ll_K (1+B+\ldots+B^d) H_K(F) \ll_K B^d H_K(F). \end{align*}$$

Using the induction hypothesis, one verifies

$$ \begin{align*} S_T(F, B) \ll_{K,n,r}& 2^{c(1)(d+1)^n}(\log H_K(F) + 1)^{10} B^{1/2}B^{r-1} \\ &+ B 2^{c(r) (d+1)^n} (\log H_K(F)+1)^{10} B^{(r-1)-1/2} \log(B)^{10(r-1)}, \end{align*} $$

where $c(r)=118+10(r-1)$ , arriving at the required bounds.

Proof. By the previous theorem, the number of $\underline {t} \in [B]_{{\mathcal {O}}_K}^r$ for a given $B \ge 1$ for which $F_i(\underline {t}, \underline {Y})$ is reducible over K is at most

$$\begin{align*}S_T(F_i, B) \ll_{K,n,r} 2^{O_{n,r}(d^n)} (\log H_K(F_i) +1)^{10}B^{r-1/2}\log(B)^{10(r-1)}. \end{align*}$$

Recall that $\# [B]_{{\mathcal {O}}_K} \ll _K B$ . Therefore, the number of $\underline {t}\in [B]_{{\mathcal {O}}_K}^r$ for which all $F_i(\underline {t}, \underline {Y})$ are irreducible over K is bounded from below by

$$\begin{align*}\#[B]_{{\mathcal{O}}_K}^r - \sum_{i=1}^m S_T(F_i, B) \gg_{K,n,r} B^r - m B^{r-1/2} \log(B)^{10(r-1)}2^{O_{n,r}(d^n)} \sum_{i=1}^m (\log H_K(F_i) +1)^{10}. \end{align*}$$

In order to guarantee we have at least one $\underline {t} \in [B]_{{\mathcal {O}}_K}^r$ , not lying on X, and such that each specialization $F_i(\underline {t},\underline {Y})$ is irreducible; it is enough to find B satisfying the following inequality:

(4.3) $$ \begin{align} \frac{B^{1/2}}{\log(B)^{10(r-1)}} &\gg_{K,n,r} m 2^{O_{n,r}(d^n)} \sum_{i=1}^m (\log H_K(F_i) +1)^{10}+ \frac{N_{\mathrm{aff}}(X,B)}{B^{r-1/2}\log(B)^{10(r-1)}}. \end{align} $$

Using the Schwartz-Zippel bound, we have $N_{\mathrm {aff}}(X,B) \ll _K s B^{r-1}$ . Note that $\frac {B^{1/2}}{\log (B)^{10(r-1)}}$ is an increasing function for $B \gg _n 1$ which is larger than a number y as soon as $B=O_r(y^2 \log ^{20(r-1)}(y))$ . Using Jensen’s inequality on $g(y)=y^2 \log ^{20(r-1)}(y)$ where y is the RHS of (4.3), we conclude the desired quantity is positive, if we demand

$$ \begin{align*} B \gg_{K,n,r} s^{2}\log(s)^{20(r-1)}+\left(m^2 2^{O_{n,r}(d^n)} \sum_{i=1}^m (\log H_K(F_i) +1)^{10} \right)^3.\\[-47pt] \end{align*} $$

Proof. We use Noether forms as in Theorem 2.5. In characteristic $0$ , we refer to [Reference Castryck, Cluckers, Dittmann and NguyenCCDN20, Lemma 4.3.7] noting that the statement and proof in [Reference Castryck, Cluckers, Dittmann and NguyenCCDN20] should be amended by correctly concluding the degree of the form F to be at most $(n-1)d(d^2-1)$ , instead of the written expression $(n-1)(d^2-1)$ . A similar correction should be made in [Reference Paredes and SasykPS22, Lemma 7.5]. See [Reference VermeulenVer22, Lemma 4.6] for $\operatorname {\mathrm {char}}(K)>0$ .

Proof. Factor $f=h_1 \cdot \ldots \cdot h_{m}$ into absolutely irreducible (homogeneous) factors and let $Z_i$ be the variety cut by $h_i$ in ${\mathbb {P}}^n$ . By Lemma 4.7, for each i there exists a hypersurface $Y_i \subset ({\mathbb {P}}^{n})^*$ of degree at most $O_n(\deg (h_i)^{C_K})$ such that if $\ell \notin Y_i$ , then $Z_i \cap H_\ell $ remains geometrically irreducible. Setting $Y=\bigcup Y_i$ , then Y is of degree at most $O_n(d^{C_K})$ .

Let g be an irreducible component of f over K and write $g=h_{i_1} \cdot \ldots \cdot h_{i_s}$ and $I_g:=\{i_1,\ldots ,i_s\}$ . Since g is irreducible over K, and by our separability assumption, it follows that $\{h_{i}\}_{i \in I_g}$ are mutually co-prime, and for each $i,j \in I_g$ there exists $\sigma _{i,j} \in \mathrm {Gal}(\overline {K}/K)$ such that $\sigma _{i,j}(Z_i)=Z_j$ (see Lemma 4.6). Set $Z_{i,j}:= Z_i \cap Z_j$ and note that $\dim Z_{i,j} = n-2$ . Consider the variety $Y^{\prime}_{i,j}:= \{ \ell \in (\mathbb {P}^{n})^* : Z_{i,j}\subseteq H_{\ell }\}$ which parameterizes hyperplanes $H_\ell $ containing $Z_{i,j}$ . Let $V_{i,j}=\mathrm {span}(Z_{i,j})$ be the $\overline {K}$ -vector space spanned by vectors in $Z_{i,j}$ . If $Z_{i,j} \subseteq H_\ell $ , then $V_{i,j} \subseteq H_\ell $ . It follows that for each $i,j\in I_g$ , the variety $Y^{\prime}_{i,j}$ can be either empty, a point, or a line according to the value of $\dim (V_{i,j})$ . Therefore, $Y_g':= \bigcup _{i,j \in I_g} Y^{\prime}_{i,j}$ contains at most $\frac {\deg (g)(\deg (g)-1)}{2}$ irreducible components, which can be lines or points. If $\ell \not \in Y^{\prime}_{i,j} \cup Y_i \cup Y_j$ , then $\dim (Z_{i,j} \cap H_\ell )=n-3$ , and therefore $h^{\prime}_{i}|_{H_{\ell }}$ and $h^{\prime}_{j}|_{H_{\ell }}$ are co-prime. It follows that if $\ell \notin Y^{\prime }_g \cup Y$ is a K-linear form, then $g|_{H_{\ell }}$ is irreducible over K, as $\sigma _{i,j}(Z_{i}\cap H_\ell )=\sigma _{i,j}(Z_{i})\cap H_\ell =Z_{j}\cap H_\ell $ for every $i,j \in I_g$ .

Write $f=g_1 \cdot \ldots \cdot g_{m'}$ with $d_i:=\deg (g_i)$ for a decomposition of f into irreducible factors over K, and let $Y':=\bigcup Y^{\prime }_{g_i}$ . Then $Y'$ contains at most $ \sum \frac {d_i(d_i-1)}{2} \le \frac {d(d-1)}{2}$ irreducible components which are either points or lines. We can thus find a hypersurface $W \supseteq Y \cup Y'$ of degree at most $O_n(d^{C_K})$ such that if $\ell \not \in W$ is a K-linear form, then each $g_i|_{H_{\ell }}$ remains irreducible over K, and therefore f is r-irreducible.

Notation 4.11. Let $f \in K[x_1, \ldots , x_n]$ be a nonconstant polynomial and let $f_i$ denote the homogenous degree i part of f. Let $I:=\{ i: f_i \neq 0\}$ . We set,

$$\begin{align*}W(f) := \bigcup_{i\in I} V(f_i)_{\operatorname{red}} \subset {\mathbb{A}}^{n}_K, \end{align*}$$

where $V(f_i)_{\operatorname {red}}$ denotes the reduced scheme cut by the radical of $(f_i)$ in ${\mathbb {A}}_K^n$ .

Proof. (1) $\Rightarrow $ (2): Suppose that f is not NCC. Then there exist distinct elements $P = (a_1:\ldots : a_n), Q = (b_1:\ldots : b_n)$ in $({\mathbb {P}}^{n-1}_K)^*(K)$ corresponding to K-linear forms $\ell _1(x):=\sum _m a_m x_m$ and $\ell _2(x):=\sum _m b_mx_m$ , and a polynomial g in $K[y_1, y_2]$ such that

$$\begin{align*}f(x_1, \ldots, x_n) = g(\ell_1(x), \ell_2(x)). \end{align*}$$

Let L be the line in $({\mathbb {P}}^{n-1}_K)^*$ through P and Q. Write $f = f_0 + f_1 + \ldots +f_d$ where $f_i$ is the homogeneous degree i part of f, and similarly for $g_i$ . Then $f_i(x) = g_i(\ell _1(x), \ell _2(x))$ , for every $i = 0, \ldots , d$ . Since every homogeneous polynomial in two variables splits over $\overline {K}$ , each $g_i$ splits over $\overline {K}$ , and we have that if $f_i$ is nonconstant then

$$\begin{align*}V(f_i)_{\operatorname{red}} = V(g_i(\ell_1(x), \ell_2(x)))_{\operatorname{red}} = H_{S_i}, \end{align*}$$

where $S_i$ is some finite subset of $L(\overline {K})$ . Hence $W(f) = H_{\cup _i S_i}$ .

(2) $\Rightarrow $ (1): For the converse, suppose that (2) holds with $P = (a_1 : \ldots : a_n), Q = (b_1: \ldots : b_n) \in ({\mathbb {P}}_K^{n-1})^*(K)$ on the line L, and $S \subset L(\overline {K})$ a finite set. Let $\ell _1(x):=\sum _m a_m x_m$ and $\ell _2(x):=\sum _m b_mx_m$ denote the corresponding K-linear forms.

Writing $f=\sum \limits _{i=0}^d f_i$ where $f_i$ is homogeneous of degree i as before, it is enough to show that for each i, we have $f_i=g_i(\ell _1(x), \ell _2(x))$ where $g_i\in K[y_1,y_2]$ . Fix i such that $f_i$ is nonconstant. Each $\ell \in S$ can be written as a $\overline {K}$ -linear combination of $\ell _1, \ell _2$ . Therefore, since $ V(f_i)_{\operatorname {red}} \subseteq W(f) =H_S=\bigcup \limits _{\ell \in S} V(\ell ), $ there exists a polynomial $h_i \in \overline {K}[y_1,y_2]$ such that $f_i(x)=h_i(\ell _1(x), \ell _2(x))$ . Write

$$\begin{align*}f_i(x)=h_i\left( \ell_1(x), \ell_2(x)\right) = \sum_j \lambda_j h_{ij}\left( \ell_1(x), \ell_2(x)\right), \end{align*}$$

where $\{\lambda _j\}_j$ forms a K-basis for $\overline {K}$ , $\lambda _1=1$ , and $h_{ij} \in K[y_1,y_2]$ . Since $f_i$ is defined over K, and $h_{ij}\left ( \ell _1(x), \ell _2(x)\right )\in K[x_1,\ldots , x_n]$ , we get by the linear independence of $\{\lambda _j\}_j$ that $h_{ij}=0$ unless $j=1$ . Therefore $h_i(y_1,y_2)=h_{i1}(y_1,y_2)$ is defined over K as required.

Proof. First note that $\delta _K(P)$ is well defined, since for any nonzero $\lambda \in \overline {K}$ the collection $\{a_{i_j}\}_j$ is linearly independent if and only if $\{\lambda a_{i_j}\}_j$ is linearly independent. For $\delta _K(P)=1$ the statement is clear; $\delta _K(P)=1$ if and only if P is defined over K. Let $s>1$ and assume by induction that the statement holds for every point $P'$ with $\delta _K(P')<s$ .

If $\delta _K(P)=s$ , we may assume without loss of generality that $\{ a_1, \ldots ,a_s\}$ forms a K-basis for $\mathrm {span}_K\{ a_1, \ldots , a_{m+1}\}$ , and write $a_j=\sum _{i=1}^{s}a_i b_{ij}$ for each j, and some $b_{ij}\in K$ . We get K-points $Q_i=(b_{i1}:\ldots :b_{i\,m+1})\in \mathbb {P}^m(K)$ , where $1 \le i \le s$ . Note that $b_{ij}=1$ if $i=j$ and $b_{ij}=0$ if $i \neq j$ and $j \le s$ . Therefore $\{Q_i\}_{i=1}^s$ are linearly independent over K, and the K-linear space $L\subseteq {\mathbb {P}}^{m}$ spanned by $Q_1,\ldots ,Q_s$ has dimension $s-1$ , and contains P. If P is contained in a different K-linear space $L'$ of dimension $s-1$ , then we have linearly independent points $Q^{\prime }_i=(b^{\prime }_{i1}:\ldots :b^{\prime }_{i\,m+1})\in L'(K)$ and we may write $a_j=\sum _{i=1}^{s}\lambda _i b^{\prime }_{ij}$ for each j and some $\lambda _i \in \overline {K}$ . Since $\{ a_1,\ldots ,a_s\} \subset \mathrm {span}_K\{ \lambda _1,\ldots ,\lambda _s\}$ is a K-basis, we may write $\lambda _j= \sum _{k=1}^s a_k\alpha _{jk}$ where $\alpha _{jk}\in K$ . Rewriting, we get for each j,

$$\begin{align*}\sum_{i=1}^{s}a_i b_{ij} =a_j =\sum_{i=1}^{s}\lambda_i b^{\prime}_{ij} =a_1\left(\sum_{k=1}^s \alpha_{k1}b^{\prime}_{kj}\right) +\ldots+a_s\left(\sum_{k=1}^s \alpha_{ks}b^{\prime}_{kj}\right). \end{align*}$$

Since $\{ a_1,\ldots ,a_s\}$ is a K-basis, we get $b_{ij}=\sum _{k=1}^s \alpha _{ki}b^{\prime }_{kj}$ for all $i,j,k$ , implying that $\{Q_1 ,\ldots ,Q_s\} \subset L'$ . It follows that $L=L'$ is unique, and that P is not contained in any K-linear space of dimension smaller than $s-1$ , as that would contradict the uniqueness of L.

If P is contained in a unique K-linear space L of dimension $s-1$ , then $\delta _K(P) \ge s$ since otherwise there exists by induction a K-linear space of dimension smaller than $s-1$ containing P, and therefore there are infinitely many K-linear spaces of dimension $s-1$ on which P lies. Since L is defined over K, there exist points $Q_1, \ldots ,Q_s\in L(K)$ , with $Q_i=(b_{i1}:\ldots :b_{i\,m+1})$ where we may assume $b_{ij} \in K$ for all $i,j $ , and scalars $\lambda _1, \ldots ,\lambda _s \in \overline {K}$ such that

$$\begin{align*}P=\lambda_1 Q_1 + \ldots +\lambda_s Q_s. \end{align*}$$

We get $a_j=\sum _{i=1}^{s}\lambda _i b_{ij}$ for each j, implying $\{ a_1,\ldots ,a_{m+1}\} \subset \mathrm {span}_K\{\lambda _1, \ldots ,\lambda _s\}$ , and therefore $\delta _K(P) \le s$ , implying $\delta _K(P)=s$ .

Proof. If f is not NCC over K, then clearly it is not NCC over $K^{\operatorname {sep}}$ .

If f is not NCC over $K^{\operatorname {sep}}$ , by Proposition 4.12 there exists a line $L \subset ({\mathbb {P}}^{n-1})^*$ defined over $K^{\operatorname {sep}}$ and a finite set $S \subset L(\overline {K})$ such that $W(f)= H_S$ . Since f is defined over K, we have that $\sigma (W(f)) = W(f)$ for all $\sigma \in \mathrm {Aut}(\overline {K}/K)$ and therefore $\sigma (S) = S$ for all $\sigma \in \mathrm {Aut}(\overline {K}/K)$ .

If $|S|\ge 2$ , since S consists of points on the line L, then the automorphism group $\mathrm {Aut}(\overline {K}/K)$ must preserve L. Since L is defined over $K^{\operatorname {sep}}$ , this implies that $\sigma (L) = L$ for every $\sigma \in \mathrm {Gal}(K^{\operatorname {sep}}/K)$ . Hence Lemma 4.6 implies that L can be defined over K, which proves the required statement using Proposition 4.12.

We may thus assume that $S = \{\ell \}$ is a singleton. Then we have that $\sigma (\ell ) = \ell $ for every $\sigma \in \mathrm {Aut}(\overline {K}/K)$ . Write $\ell = (c_1 : \ldots : c_n)$ , then without loss of generality we may assume that $c_1 = 1$ . It follows that $\sigma (c_i) = c_i$ for every $\sigma \in \mathrm {Aut}(\overline {K}/K)$ .

Consider now $\delta _{K^{\operatorname {sep}}}(\ell )$ as in Proposition 4.13. Since $\ell $ lies on the $K^{\operatorname {sep}}$ -line L, then $\delta _{K^{\operatorname {sep}}}(\ell ) \le 2$ . If $\delta _{K^{\operatorname {sep}}}(\ell )=1$ , then $\ell $ is defined over $K^{\operatorname {sep}}$ , and therefore $\ell $ is defined over K, since every element of $\mathrm {Gal}(K^{\operatorname {sep}}/K)$ fixes $\ell $ . We may thus take any line defined over K which passes through $\ell $ to deduce Condition (2) of Proposition 4.12.

Assume $\delta _{K^{\operatorname {sep}}}(\ell )=2$ . By Proposition 4.13, there exists a unique line L defined over $K^{\operatorname {sep}}$ which contains $\ell $ . Since for each $\sigma \in \mathrm {Aut}(\overline {K}/K)$ we have $\sigma (\ell )=\ell $ , the line $\sigma (L)$ is defined over $K^{\operatorname {sep}}$ and contains $\ell $ . We get $\sigma (L)=L$ for every $\sigma \in \mathrm {Aut}(\overline {K}/K)$ , and thus L is defined over K by Lemma 4.6.

Proof. If f is not NCC, then clearly $\dim _K V \leq 2$ .

Conversely, assume that $\dim _K V \leq 2$ . Without loss of generality, we can assume that $\frac {\partial f}{\partial x_1}, \frac {\partial f}{\partial x_2}$ span V. For each i there are $a_i, b_i$ in K such that

$$\begin{align*}\frac{\partial f}{\partial x_i} = a_i \frac{\partial f}{\partial x_1} + b_i \frac{\partial f}{\partial x_2}. \end{align*}$$

Consider the polynomial

$$\begin{align*}g(y_1, \ldots, y_n) = f\left( y_1 - \sum_{i\geq 3} a_i y_i, y_2 - \sum_{i\geq 3} b_i y_i, y_3, \ldots, y_n\right) \in K[y_1, \ldots, y_n]. \end{align*}$$

Then $\frac {\partial g}{\partial y_i} = 0$ whenever $i\geq 3$ , and so $g(y)$ is a polynomial only involving $y_1, y_2$ . Hence f is not NCC.

Proof. First note that $f_d|_{V(\ell )} = 0$ if and only if $\ell $ is a factor of $f_d$ . There are thus at most d linear forms $\ell $ such that $f_d|_{V(\ell )} = 0$ , which we avoid in the proof below.

(1). We use Proposition 4.12. If $W(f)$ contains a geometrically irreducible component Z of degree at least $2$ , by Lemma 4.7, there exists a form $F \in {\mathbb {Z}}[x_1,\ldots ,x_n]$ in the coefficients of $\ell $ of degree at most $O_n(d^{C_K})$ , such that if $F(\ell ) \neq 0$ , then $Z \cap V(\ell )$ is geometrically irreducible, and therefore $f|_{V(\ell )}$ is NCC over K.

Assume now that $W(f)$ contains at least three hyperplanes whose corresponding linear forms are linearly independent, and denote the linear space they span in $({\mathbb {P}}^{n-1}_K)^*$ by V. Then $W(f|_{V(\ell )})$ contains three such hyperplanes for any K-linear form $\ell $ which does not lie on V, and for which $f_i|_{V(\ell )}\neq 0$ for indices i where $W(f_i)$ contains one of the hyperplanes as above. It follows by Proposition 4.12 that $f|_{V(\ell )}$ is NCC over K in this case. We may thus find a hypersurface of degree at most $O(d)$ in $({\mathbb {P}}^{n-1}_K)^*$ such that $f|_{V(\ell )}$ is NCC over K for every $\ell $ not lying on it.

Finally, assume that $W(f)=H_S$ , where S is a finite set contained in a $\overline {K}$ -line $L \subseteq ({\mathbb {P}}^{n-1}_K)^*$ which is not defined over K. By Proposition 4.14, we may assume K is separably closed and therefore if $\operatorname {\mathrm {char}}(K)=0$ we are done by Proposition 4.12. It follows by Proposition 4.13 that there exists a hyperplane $V(\ell ') \subset W(f)$ , corresponding to a point $\ell '=(a_1:\ldots :a_n)\in ({\mathbb {P}}^{n-1}_{K})^*$ , satisfying $\delta _K(\ell ')\ge 2$ . If $\delta _K(\ell ')\ge 3$ , then without loss of generality we may assume that $a_1, a_2, a_3$ are linearly independent over K. We may thus choose a K-linear form $\ell (x)=\sum b_i x_i$ of height at most $O(d)$ such that $\delta _K(\ell '|_{V(\ell )})\geq 3$ by taking $b_i \neq 0$ for any $i \geq 4$ . In particular, any $\ell $ not lying on the hypersurface cut in $({\mathbb {P}}^{n-1}_{K})^*$ by $\prod b_i =0$ is as desired. We then get that $ V(\ell ')\cap {V(\ell )} \subset W(f|_{V(\ell )})$ , considered as a point in $({\mathbb {P}}_K^{n-2})^*$ , does not lie on any line defined over K by Proposition 4.13, and therefore $f|_{V(\ell )}$ is NCC. We may thus assume that $\delta _K(\ell ') \le 2$ for every hyperplane $V(\ell ') \subset W(f)$ . Fix hyperplanes $H_{\ell _1}, H_{\ell _2} \subset W(f)$ with $\delta _K(\ell _1)=2$ and such that $\ell _2$ does not lie on the unique K-line $L'\subset ({\mathbb {P}}^{n-1}_K)^*$ containing $\ell _1$ . Note that $ \ell _2 \notin L\subset ({\mathbb {P}}^{n-1}_K)^*$ if and only if $V_{L'}:=\{\ell _1'(x)=\ell _2'(x)=0\} \not \subset H_{\ell _2}$ , where $\ell _1'$ , $ \ell _2'$ are two linearly independent K-linear forms lying on $L'$ . We may thus choose any $\ell $ such that the conditions $H_{\ell _2} \cap V_{L'} \not \subset V(\ell )$ , $\delta _K({\ell _1}|_{V(\ell )})=2$ and $f_d|_{V(\ell )} \neq 0$ hold. Therefore, noting that $\dim (H_{\ell _2} \cap V_{L'})=n-3$ and using arguments similar to the $\delta _K \ge 3$ case, we may find a hypersurface in $({\mathbb {P}}^{n-1}_K)^*$ of degree at most $O_n(d)$ such that every linear form not lying on it satisfies our desired condition. The final claim follows from the Schwartz–Zippel bound. (2). If $W(f)$ contains a nonlinear geometrically irreducible component or at least three hyperplanes whose corresponding linear forms are linearly independent, we may follow the proof of (1) with the linear forms $\tilde {\ell }_a(x):=\ell _1(x) + a\ell _2(x), a\in K$ , noting that $f|_{V(\tilde {\ell }_0)}$ is NCC. We may then bound the number of values of a for which $f|_{V(\tilde {\ell }_a)}$ is not NCC by $O_n(d^{C_K'})$ with $C_K'$ as in (1). If we are not in one of the above two situations, then $W(f)=H_S$ where $S \subset L$ is a finite set contained in a line $L \subseteq ({\mathbb {P}}^{n-1}_{K})^*$ not defined over K. In particular, each homogeneous part $f_i$ of f splits into linear factors. Assume towards a contradiction that $f|_{V(\ell _1+a_i\ell _2)}$ is not NCC for distinct values $a_1 \neq a_2$ in K. Then there exist K-lines $L_i \subset (\mathbb {P}^{n-2}_K)^*$ such that $W(f|_{V(\ell _1+a_i\ell _2)})=H_{S_i}$ and ${S_i} \subset L_i$ for $i=1,2$ . It follows that $S \subset P_i$ , where $P_i$ is the K-plane in $({\mathbb {P}}^{n-1}_{K})^*$ containing the preimage of $L_i$ under the projection