Recovering an Homogeneous Polynomial from Moments of Its Level Set

Let $\mathbf{K }:=\left\{ \mathbf{x }: g(\mathbf{x })\le 1\right\} $ K : = x : g ( x ) ≤ 1 be the compact (and not necessarily convex) sub-level set of some homogeneous polynomial $g$ g . Assume that the only knowledge about $\mathbf{K }$ K is the degree of $g$ g as well as the moments of the Lebesgue measure on $\mathbf{K }$ K up to order $2d$ 2 d . Then the vector of coefficients of $g$ g is the solution of a simple linear system whose associated matrix is nonsingular. In other words, the moments up to order $2d$ 2 d of the Lebesgue measure on $\mathbf{K }$ K encode all information on the homogeneous polynomial $g$ g that defines $\mathbf{K }$ K (in fact, only moments of order $d$ d and $2d$ 2 d are needed).     

Algebraic Boundaries of \mathrm{SO}(2)-Orbitopes

Let $X\subset \mathbb{A }^{2r}$ X ? A 2 r be a real curve embedded into an even-dimensional affine space. We characterise when the $r$ r th secant variety to $X$ X is an irreducible component of the algebraic boundary of the convex hull of the real points $X(\mathbb{R })$ X ( R ) of $X$ X . This fact is then applied to $4$ 4 -dimensional $\mathrm{SO}(2)$ SO ( 2 ) -orbitopes and to the so called Barvinok–Novik orbitopes to study when they are basic closed semi-algebraic sets. In the case of $4$ 4 -dimensional $\mathrm{SO}(2)$ SO ( 2 ) -orbitopes, we find all irreducible components of their algebraic boundary.     

Maximal Equilateral Sets

A subset of a normed space $X$ X is called equilateral if the distance between any two points is the same. Let $m(X)$ m ( X ) be the smallest possible size of an equilateral subset of $X$ X maximal with respect to inclusion. We first observe that Petty’s construction of a $d$ d - $X$ X of any finite dimension $d\ge 4$ d ≥ 4 with $m(X)=4$ m ( X ) = 4 can be generalised to give $m(X\oplus _1\mathbb R )=4$ m ( X ⊕ 1 R ) = 4 for any $X$ X of dimension at least $2$ 2 which has a smooth point on its unit sphere. By a construction involving Hadamard matrices we then show that for any set $\Gamma $ Γ , $m(\ell _p(\Gamma ))$ m ( ? p ( Γ ) ) is finite and bounded above by a function of $p$ p , for all $1\le p<2$ 1 ≤ p < 2 . Also, for all $p\in [1,\infty )$ p ∈ [ 1 , ∞ ) and $d\in \mathbb N $ d ∈ N there exists $c=c(p,d)>1$ c = c ( p , d ) > 1 such that $m(X)\le d+1$ m ( X ) ≤ d + 1 for all $d$ d - $X$ X with Banach–Mazur distance less than $c$ c from $\ell _p^d$ ? p d . Using Brouwer’s fixed-point theorem we show that $m(X)\le d+1$ m ( X ) ≤ d + 1 for all $d$ d - $X$ X with Banach–Mazur distance less than $3/2$ 3 / 2 from $\ell _\infty ^d$ ? ∞ d . A graph-theoretical argument furthermore shows that $m(\ell _\infty ^d)=d+1$ m ( ? ∞ d ) = d + 1 . The above results lead us to conjecture that $m(X)\le 1+\dim X$ m ( X ) ≤ 1 + dim X for all finite-normed spaces $X$ X .     

Approximating Tverberg Points in Linear Time for Any Fixed Dimension

Let $P \subseteq \mathbb{R }^d$ P ? R d be a $d$ d -dimensional $n$ n -point set. A Tverberg partition is a partition of $P$ P into $r$ r sets $P_1, \dots , P_r$ P 1 , ? , P r such that the convex hulls $\hbox {conv}(P_1), \dots , \hbox {conv}(P_r)$ conv ( P 1 ) , ? , conv ( P r ) have non-empty intersection. A point in $\bigcap _{i=1}^{r} \hbox {conv}(P_i)$ ? i = 1 r conv ( P i ) is called a Tverberg point of depth $r$ r for $P$ P . A classic result by Tverberg shows that there always exists a Tverberg partition of size $\lceil n/(d+1) \rceil $ ? n / ( d + 1 ) ? , but it is not known how to find such a partition in polynomial time. Therefore, approximate solutions are of interest. We describe a deterministic algorithm that finds a Tverberg partition of size $\lceil n/4(d+1)^3 \rceil $ ? n / 4 ( d + 1 ) 3 ? in time $d^{O(\log d)} n$ d O ( log d ) n . This means that for every fixed dimension we can compute an approximate Tverberg point (and hence also an approximate centerpoint) in linear time. Our algorithm is obtained by combining a novel lifting approach with a recent result by Miller and Sheehy (Comput Geom Theory Appl 43(8):647–654, 2010).     

Triangle-Free Geometric Intersection Graphs with Large Chromatic Number

Several classical constructions illustrate the fact that the chromatic number of a graph may be arbitrarily large compared to its clique number. However, until very recently no such construction was known for intersection graphs of geometric objects in the plane. We provide a general construction that for any arc-connected compact set $X$ X in $\mathbb{R }^2$ R 2 that is not an axis-aligned rectangle and for any positive integer $k$ k produces a family $\mathcal{F }$ F of sets, each obtained by an independent horizontal and vertical scaling and translation of $X$ X , such that no three sets in $\mathcal{F }$ F pairwise intersect and $\chi (\mathcal{F })>k$ χ ( F ) > k . This provides a negative answer to a question of Gyárfás and Lehel for L-shapes. With extra conditions we also show how to construct a triangle-free family of homothetic (uniformly scaled) copies of a set with arbitrarily large chromatic number. This applies to many common shapes, like circles, square boundaries or equilateral L-shapes. Additionally, we reveal a surprising connection between coloring geometric objects in the plane and on-line coloring of intervals on the line.     

Bounds on the Complexity of Halfspace Intersections when the Bounded Faces have Small Dimension

For a polyhedron $P$ P let $B(P)$ B ( P ) denote the polytopal complex that is formed by all bounded faces of $P$ P . If $P$ P is the intersection of $n$ n halfspaces in $\mathbb R ^D$ R D , but the maximum dimension $d$ d of any face in $B(P)$ B ( P ) is much smaller, we show that the combinatorial complexity of $P$ P cannot be too high; in particular, that it is independent of $D$ D . We show that the number of vertices of $P$ P is $O(n^d)$ O ( n d ) and the total number of bounded faces of the polyhedron is $O(n^{d^2})$ O ( n d 2 ) . For inputs in general position the number of bounded faces is $O(n^d)$ O ( n d ) . We show that for certain specific values of $d$ d and $D$ D , our bounds are tight. For any fixed $d$ d , we show how to compute the set of all vertices, how to determine the maximum dimension of a bounded face of the polyhedron, and how to compute the set of bounded faces in polynomial time, by solving a number of linear programs that is polynomial in  $n$ n .     

Tilings of Parallelograms with Similar Right Triangles

We say that a triangle $T$ T tiles the polygon $\mathcal A $ A if $\mathcal A $ A can be decomposed into finitely many non-overlapping triangles similar to $T$ T . A tiling is called regular if there are two angles of the triangles, say $\alpha $ α and $\beta $ β , such that at each vertex $V$ V of the tiling the number of triangles having $V$ V as a vertex and having angle $\alpha $ α at $V$ V is the same as the number of triangles having angle $\beta $ β at $V$ V . Otherwise the tiling is called irregular. Let $\mathcal P (\delta )$ P ( δ ) be a parallelogram with acute angle $\delta $ δ . In this paper we prove that if the parallelogram $\mathcal P (\delta )$ P ( δ ) is tiled with similar triangles of angles $(\alpha , \beta , \pi /2)$ ( α , β , π / 2 ) , then $(\alpha , \beta )=(\delta , \pi /2-\delta )$ ( α , β ) = ( δ , π / 2 - δ ) or $(\alpha , \beta )=(\delta /2, \pi /2-\delta /2)$ ( α , β ) = ( δ / 2 , π / 2 - δ / 2 ) , and if the tiling is regular, then only the first case can occur.     

The Funk and Hilbert geometries for spaces of constant curvature

The goal of this paper is to introduce and to study analogues of the Euclidean Funk and Hilbert metrics on open convex subsets of the hyperbolic space $\mathbb H ^n$ H n and of the sphere $S^n$ S n . We highlight some striking similarities among the three cases (Euclidean, spherical and hyperbolic) which hold at least at a formal level. The proofs of the basic properties of the classical Funk metric on subsets of $\mathbb R ^n$ R n use similarity properties of Euclidean triangles which of course do not hold in the non-Euclidean cases. Transforming the side lengths of triangles using hyperbolic and circular functions and using some non-Euclidean trigonometric formulae, the Euclidean similarity techniques are transported into the non-Euclidean worlds. We start by giving three representations of the Funk metric in each of the non-Euclidean cases, which parallel known representations for the Euclidean case. The non-Euclidean Funk metrics are shown to be Finslerian, and the associated Finsler norms are described. We then study their geodesics. The Hilbert geometry of convex sets in the non-Euclidean constant curvature spaces $S^n$ S n and $\mathbb H ^n$ H n is then developed by using the properties of the Funk metric and by introducing a non-Euclidean cross ratio. In the case of Euclidean (respectively spherical, hyperbolic) geometry, the Euclidean (respectively spherical, hyperbolic) geodesics are Funk and Hilbert geodesics. This leads to a formulation and a discussion of Hilbert’s Problem IV in the non-Euclidean settings. Projection maps between the spaces $\mathbb R ^n, \mathbb H ^n$ R n , H n and the upper hemisphere establish equivalences between the Hilbert geometries of convex sets in the three spaces of constant curvature, but such an equivalence does not hold for Funk geometries.     

Boundaries of Disk-Like Self-affine Tiles

Let $T:= T(A, \mathcal{D })$ T : = T ( A , D ) be a disk-like self-affine tile generated by an integral expanding matrix $A$ A and a consecutive collinear digit set $\mathcal{D }$ D , and let $f(x)=x^{2}+px+q$ f ( x ) = x 2 + px + q be the characteristic polynomial of $A$ A . In the paper, we identify the boundary $\partial T$ ? T with a sofic system by constructing a neighbor graph and derive equivalent conditions for the pair $(A,\mathcal{D })$ ( A , D ) to be a number system. Moreover, by using the graph-directed construction and a device of pseudo-norm $\omega $ ω , we find the generalized Hausdorff dimension $\dim _H^{\omega } (\partial T)=2\log \rho (M)/\log |q|$ dim H ω ( ? T ) = 2 log ρ ( M ) / log | q | where $\rho (M)$ ρ ( M ) is the spectral radius of certain contact matrix $M$ M . Especially, when $A$ A is a similarity, we obtain the standard Hausdorff dimension $\dim _H (\partial T)=2\log \rho /\log |q|$ dim H ( ? T ) = 2 log ρ / log | q | where $\rho $ ρ is the largest positive zero of the cubic polynomial $x^{3}-(|p|-1)x^{2}-(|q|-|p|)x-|q|$ x 3 ? ( | p | ? 1 ) x 2 ? ( | q | ? | p | ) x ? | q | , which is simpler than the known result.     

Solution of the Propeller Conjecture in \mathbb R ^3

It is shown that every measurable partition $\{A_1,\ldots , A_k\}$ { A 1 , … , A k } of $\mathbb R ^3$ R 3 satisfies 1 $$\begin{aligned} \sum _{i=1}^k\big \Vert \int _{A_i} x\mathrm{{e}}^{-\frac{1}{2}\Vert x\Vert _2^2}\mathrm{{d}}x\big \Vert _2^2\leqslant 9\pi ^2. \end{aligned}$$ ∑ i = 1 k ‖ ∫ A i x e - 1 2 ‖ x ‖ 2 2 d x ‖ 2 2 ? 9 π 2 . Let $\{P_1,P_2,P_3\}$ { P 1 , P 2 , P 3 } be the partition of $\mathbb R ^2$ R 2 into $120^{\circ }$ 120 ° sectors centered at the origin. The bound (1) is sharp, with equality holding if $A_i=P_i\times \mathbb R $ A i = P i × R for $i\in \{1,2,3\}$ i ∈ { 1 , 2 , 3 } and $A_i=\emptyset $ A i = ? for $i\in \{4,\ldots ,k\}$ i ∈ { 4 , … , k } . This settles positively the $3$ 3 -dimensional Propeller Conjecture of Khot and Naor [(Mathematika 55(1-2):129–165, 2009 (FOCS 2008)]. The proof of (1) reduces the problem to a finite set of numerical inequalities which are then verified with full rigor in a computer-assisted fashion. The main consequence (and motivation) of (1) is complexity-theoretic: the unique games hardness threshold of the kernel clustering problem with $4\times 4$ 4 × 4 centered and spherical hypothesis matrix equals $\frac{2\pi }{3}$ 2 π 3 .     

Appell Bases for Monogenic Functions of Three Variables

Monogenic (or hyperholomorphic) functions are well known in general Clifford algebras but have been little studied in the particular case ${\mathbb{R}^{3} \rightarrow \mathbb{R}^{3}}$ R 3 → R 3 . We describe for this case the collection of all Appell systems: bases for the finite-dimensional spaces of monogenic homogeneous polynomials which respect the operator ${D = \partial_{0} - \vec{\partial}}$ D = ? 0 ? ? → . We prove that no purely algebraic recursive formula (in a specific sense) exists for these Appell systems, in contrast to the existence of known constructions for ${\mathbb{R}^{3} \rightarrow \mathbb{R}^{4}}$ R 3 → R 4 and ${\mathbb{R}^{4} \rightarrow \mathbb{R}^{4}}$ R 4 → R 4 . However, we give a simple recursive procedure for constructing Appell bases for ${\mathbb{R}^{3} \rightarrow \mathbb{R}^{3}}$ R 3 → R 3 which uses the operation of integration of polynomials.     

On multiplicative (generalized)-derivations in prime and semiprime rings

Let R be a ring. A map ${F : R \rightarrow R}$ F : R → R is called a multiplicative (generalized)-derivation if F(xy) = F(x)y + xg(y) is fulfilled for all ${x, y \in R}$ x , y ∈ R where ${g : R \rightarrow R}$ g : R → R is any map (not necessarily derivation). The main objective of the present paper is to study the following situations: (i) ${F(xy) \pm xy \in Z}$ F ( xy ) ± xy ∈ Z , (ii) ${F(xy) \pm yx \in Z}$ F ( xy ) ± yx ∈ Z , (iii) ${F(x)F(y) \pm xy \in Z}$ F ( x ) F ( y ) ± xy ∈ Z and (iv) ${F(x)F(y) \pm yx \in Z}$ F ( x ) F ( y ) ± yx ∈ Z for all x, y in some appropriate subset of R. Moreover, some examples are also given.     

Characterization of Co-blockers for Simple Perfect Matchings in a Convex Geometric Graph

Consider the complete convex geometric graph on $2m$ 2 m vertices, CGG $(2m)$ ( 2 m ) , i.e., the set of all boundary edges and diagonals of a planar convex $2m$ 2 m -gon P. In (Keller and Perles, Israel J Math 187:465–484, 2012), the smallest sets of edges that meet all the simple perfect matchings (SPMs) in CGG $(2m)$ ( 2 m ) (called “blockers”) are characterized, and it is shown that all these sets are caterpillar graphs with a special structure, and that their total number is $m \cdot 2^{m-1}$ m · 2 m ? 1 . In this paper we characterize the co-blockers for SPMs in CGG $(2m)$ ( 2 m ) , that is, the smallest sets of edges that meet all the blockers. We show that the co-blockers are exactly those perfect matchings M in CGG $(2m)$ ( 2 m ) where all edges are of odd order, and two edges of M that emanate from two adjacent vertices of P never cross. In particular, while the number of SPMs and the number of blockers grow exponentially with m, the number of co-blockers grows super-exponentially.     

*-Quantizations of Fourier–Mukai Transforms

We study deformations of Fourier–Mukai transforms in general complex analytic settings. Suppose X and Y are complex manifolds, and let P be a coherent sheaf on X ×  Y. Suppose that the Fourier–Mukai transform ${\Phi}$ Φ given by the kernel P is an equivalence between the coherent derived categories of X and of Y. Suppose also that we are given a formal *-quantization ${\mathbb{X}}$ X of X. Our main result is that ${\mathbb{X}}$ X gives rise to a unique formal *-quantization ${\mathbb{Y}}$ Y of Y. For the statement to hold, *-quantizations must be understood in the framework of stacks of algebroids. The quantization ${\mathbb{Y}}$ Y is uniquely determined by the condition that ${\Phi}$ Φ deforms to an equivalence between the derived categories of ${\mathbb{X}}$ X and ${\mathbb{Y}}$ Y . Equivalently, the condition is that P deforms to a coherent sheaf ${\tilde{P}}$ P ~ on the formal *-quantization ${\mathbb{X} \times\mathbb{Y}^{op}}$ X × Y o p of X × Y; here ${\mathbb{Y}^{op}}$ Y o p is the opposite of the quantization ${\mathbb{Y}}$ Y .     

A Reduction Theorem for the Kripke–Joyal Semantics: Forcing Over an Arbitrary Category can Always be Replaced by Forcing Over a Complete Heyting Algebra

It is assumed that a Kripke–Joyal semantics ${\mathcal{A} = \left\langle \mathbb{C},{\rm Cov}, {\it F},\Vdash \right\rangle}$ A = C , Cov , F , ? has been defined for a first-order language ${\mathcal{L}}$ L . To transform ${\mathbb{C}}$ C into a Heyting algebra ${\overline{\mathbb{C}}}$ C ¯ on which the forcing relation is preserved, a standard construction is used to obtain a complete Heyting algebra made up of cribles of ${\mathbb{C}}$ C . A pretopology ${\overline{{\rm Cov}}}$ Cov ¯ is defined on ${\overline{\mathbb{C}}}$ C ¯ using the pretopology on ${\mathbb{C}}$ C . A sheaf ${\overline{{\it F}}}$ F ¯ is made up of sections of F that obey functoriality. A forcing relation ${\overline{\Vdash}}$ ? ¯ is defined and it is shown that ${\overline{\mathcal{A}} = \left\langle \overline{\mathbb{C}},\overline{\rm{Cov}},\overline{{\it F}}, \overline{\Vdash} \right\rangle }$ A ¯ = C ¯ , Cov ¯ , F ¯ , ? ¯ is a Kripke–Joyal semantics that faithfully preserves the notion of forcing of ${\mathcal{A}}$ A . That is to say, an object a of ${\mathbb{C}Ob}$ C O b forces a sentence with respect to ${\mathcal{A}}$ A if and only if the maximal a-crible forces it with respect to ${\overline{\mathcal{A}}}$ A ¯ . This reduces a Kripke–Joyal semantics defined over an arbitrary site to a Kripke–Joyal semantics defined over a site which is based on a complete Heyting algebra.     

Improved Enumeration of Simple Topological Graphs

A simple topological graph $T=(V(T), E(T))$ T = ( V ( T ) , E ( T ) ) is a drawing of a graph in the plane where every two edges have at most one common point (an endpoint or a crossing) and no three edges pass through a single crossing. Topological graphs $G$ G and $H$ H are isomorphic if $H$ H can be obtained from $G$ G by a homeomorphism of the sphere, and weakly isomorphic if $G$ G and $H$ H have the same set of pairs of crossing edges. We generalize results of Pach and Tóth and the author’s previous results on counting different drawings of a graph under both notions of isomorphism. We prove that for every graph $G$ G with $n$ n vertices, $m$ m edges and no isolated vertices the number of weak isomorphism classes of simple topological graphs that realize $G$ G is at most $2^{O(n^2\log (m/n))}$ 2 O ( n 2 log ( m / n ) ) , and at most $2^{O(mn^{1/2}\log n)}$ 2 O ( m n 1 / 2 log n ) if $m\le n^{3/2}$ m ≤ n 3 / 2 . As a consequence we obtain a new upper bound $2^{O(n^{3/2}\log n)}$ 2 O ( n 3 / 2 log n ) on the number of intersection graphs of $n$ n pseudosegments. We improve the upper bound on the number of weak isomorphism classes of simple complete topological graphs with $n$ n vertices to $2^{n^2\cdot \alpha (n)^{O(1)}}$ 2 n 2 · α ( n ) O ( 1 ) , using an upper bound on the size of a set of permutations with bounded VC-dimension recently proved by Cibulka and the author. We show that the number of isomorphism classes of simple topological graphs that realize $G$ G is at most $2^{m^2+O(mn)}$ 2 m 2 + O ( m n ) and at least $2^{\Omega (m^2)}$ 2 Ω ( m 2 ) for graphs with $m>(6+\varepsilon )n$ m > ( 6 + ε ) n .     

The 2-Page Crossing Number of K_{n}

Around 1958, Hill described how to draw the complete graph $K_n$ K n with $$\begin{aligned} Z(n) :=\frac{1}{4}\Big \lfloor \frac{n}{2}\Big \rfloor \Big \lfloor \frac{n-1}{2}\Big \rfloor \Big \lfloor \frac{n-2}{2}\Big \rfloor \Big \lfloor \frac{n-3}{2}\Big \rfloor \end{aligned}$$ Z ( n ) : = 1 4 ? n 2 ? ? n ? 1 2 ? ? n ? 2 2 ? ? n ? 3 2 ? crossings, and conjectured that the crossing number ${{\mathrm{cr}}}(K_{n})$ cr ( K n ) of $K_n$ K n is exactly $Z(n)$ Z ( n ) . This is also known as Guy’s conjecture as he later popularized it. Towards the end of the century, substantially different drawings of $K_{n}$ K n with $Z(n)$ Z ( n ) crossings were found. These drawings are 2-page book drawings, that is, drawings where all the vertices are on a line $\ell $ ? (the spine) and each edge is fully contained in one of the two half-planes (pages) defined by  $\ell $ ? . The 2-page crossing number of $K_{n} $ K n , denoted by $\nu _{2}(K_{n})$ ν 2 ( K n ) , is the minimum number of crossings determined by a 2-page book drawing of $K_{n}$ K n . Since ${{\mathrm{cr}}}(K_{n}) \le \nu _{2}(K_{n})$ cr ( K n ) ≤ ν 2 ( K n ) and $\nu _{2}(K_{n}) \le Z(n)$ ν 2 ( K n ) ≤ Z ( n ) , a natural step towards Hill’s Conjecture is the weaker conjecture $\nu _{2}(K_{n}) = Z(n)$ ν 2 ( K n ) = Z ( n ) , popularized by Vrt’o. In this paper we develop a new technique to investigate crossings in drawings of $K_{n}$ K n , and use it to prove that $\nu _{2}(K_{n}) = Z(n) $ ν 2 ( K n ) = Z ( n ) . To this end, we extend the inherent geometric definition of $k$ k -edges for finite sets of points in the plane to topological drawings of $K_{n}$ K n . We also introduce the concept of ${\le }{\le }k$ ≤ ≤ k -edges as a useful generalization of ${\le }k$ ≤ k -edges and extend a powerful theorem that expresses the number of crossings in a rectilinear drawing of $K_{n}$ K n in terms of its number of ${\le }k$ ≤ k -edges to the topological setting. Finally, we give a complete characterization of crossing minimal 2-page book drawings of $K_{n}$ K n and show that, up to equivalence, they are unique for $n$ n even, but that there exist an exponential number of non homeomorphic such drawings for $n$ n odd.     

A Question from a Famous Paper of Erdős

Given a convex body $K$ K , consider the smallest number $N$ N so that there is a point $P\in \partial K$ P ∈ ? K such that every circle centred at $P$ P intersects $\partial K$ ? K in at most $N$ N points. In 1946 Erd?s conjectured that $N=2$ N = 2 for all $K$ K , but there are convex bodies for which this is not the case. As far as we know there is no known global upper bound. We show that no convex body has $N=\infty $ N = ∞ and that there are convex bodies for which $N = 6$ N = 6 .     

Many Empty Triangles have a Common Edge

Given a finite point set $X$ X in the plane, the degree of a pair $\{x,y\} \subset X$ { x , y } ? X is the number of empty triangles $t=\mathrm {conv} \{x,y,z\},$ t = conv { x , y , z } , where empty means $t\cap X=\{x,y,z\}.$ t ∩ X = { x , y , z } . Define $\deg X$ deg X as the maximal degree of a pair in $X.$ X . Our main result is that if $X$ X is a random sample of $n$ n independent and uniform points from a fixed convex body, then $\deg X \ge cn/\ln n$ deg X ≥ cn / ln n in expectation.     

Nef line bundles on flag bundles on a curve over {\overline{{\mathbb F}}_p}

Let E be a vector bundle of rank r over an irreducible smooth projective curve X defined over the field ${\overline{{\mathbb F}}_p}$ F ¯ p . For fixed integers ${r_1\, , \ldots\, , r_\nu}$ r 1 , ... , r ν with ${1\, \leq\, r_1\, <\, \cdots\, <\, r_\nu\, <\, r}$ 1 ≤ r 1 < ? < r ν < r , let ${\text{Fl}(E)}$ Fl ( E ) be the corresponding flag bundle over X associated to E. Let ${\xi\, \longrightarrow \, {\rm Fl}(E)}$ ξ ? Fl ( E ) be a line bundle such that for every pair of the form ${(C\, ,\phi)}$ ( C , ? ) , where C is an irreducible smooth projective curve defined over ${\overline{\mathbb F}_p}$ F ¯ p and ${\phi\, :\, C\, \longrightarrow\, {\rm Fl}(E)}$ ? : C ? Fl ( E ) is a nonconstant morphism, the inequality ${{\rm degree}(\phi^* \xi)\, > \, 0}$ degree ( ? ? ξ ) > 0 holds. We prove that the line bundle ${\xi}$ ξ is ample.