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Shamil Asgarli

Publications and source records attributed to Shamil Asgarli.

At least 37 records · Page 2Linked to original sources
arXiv2024

Linear system of hypersurfaces passing through a Galois orbit

Let $d$ and $n$ be positive integers, and $E/F$ be a separable field extension of degree $m=\binom{n+d}{n}$. We show that if $|F| > 2$, then there exists a point $P\in \mathbb{P}^n(E)$ which does not lie on any degree $d$ hypersurface defined over $F$. In other words, the $m$ Galois conjugates of $P$ impose independent conditions on the $m$-dimensional $F$-vector space of degree $d$ forms in $x_0, x_1, \ldots, x_n$. As an application, we determine the maximal dimensions of linear systems $\mathcal{L}_1$ and $\mathcal{L}_2$ of hypersurfaces in $\mathbb P^n$ over a finite field $F$, where every $F$-member of $\mathcal{L}_1$ is reducible and every $F$-member of $\mathcal{L}_2$ is irreducible.

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arXiv2024

The subspace structure of maximum cliques in pseudo-Paley graphs from unions of cyclotomic classes

Blokhuis showed that all maximum cliques in Paley graphs of square order have a subfield structure. Recently, it has been shown that in Peisert-type graphs, all maximum cliques are affine subspaces, and yet some maximum cliques do not arise from a subfield. In this paper, we investigate the existence of a clique of size $\sqrt{q}$ with a subspace structure in pseudo-Paley graphs of order $q$ from unions of semi-primitive cyclotomic classes. We show that such a clique must have an equal contribution from each cyclotomic class and that most such pseudo-Paley graphs do not admit such cliques, suggesting that the Delsarte bound $\sqrt{q}$ on the clique number can be improved in general. We also prove that generalized Peisert graphs are not isomorphic to Paley graphs or Peisert graphs, confirming a conjecture of Mullin.

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arXiv2024

Transverse linear subspaces to hypersurfaces over finite fields

Ballico proved that a smooth projective variety $X$ of degree $d$ over a finite field of $q$ elements admits a smooth hyperplane section if $q\geq d(d-1)^{\dim X}$. In this paper, we refine this criterion for higher codimensional linear sections on smooth hypersurfaces and for hyperplane sections on Frobenius classical hypersurfaces. We also prove a similar result for the existence of reduced hyperplane sections on reduced hypersurfaces.

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arXiv2024

Most plane curves over finite fields are not blocking

A plane curve $C\subset\mathbb{P}^2$ of degree $d$ is called \emph{blocking} if every $\mathbb{F}_q$-line in the plane meets $C$ at some $\mathbb{F}_q$-point. We prove that the proportion of blocking curves among those of degree $d$ is $o(1)$ when $d\geq 2q-1$ and $q \to \infty$. We also show that the same conclusion holds for smooth curves under the somewhat weaker condition $d\geq 3p$ and $d, q \to \infty$. Moreover, the two events in which a random plane curve is smooth and respectively blocking are shown to be asymptotically independent. Extending a classical result on the number of $\mathbb{F}_q$-roots of random polynomials, we find that the limiting distribution of the number of $\mathbb{F}_q$-points in the intersection of a random plane curve and a fixed $\mathbb{F}_q$-line is Poisson with mean $1$. We also present an explicit formula for the proportion of blocking curves involving statistics on the number of $\mathbb{F}_q$-points contained in a union of $k$ lines for $k=1, 2, \ldots, q^2+q+1$.

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arXiv2023

Existence of pencils with nonblocking hypersurfaces

We prove that there is a pencil of hypersurfaces in $\mathbb{P}^n$ of any given degree over a finite field $\mathbb{F}_q$ such that every $\mathbb{F}_q$-member of the pencil is not blocking with respect to $\mathbb{F}_q$-lines.

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arXiv2023

A Fair Shake: How close can the sum of $n$-sided dice be to a uniform distribution?

Two possibly unfair $n$-sided dice, both labelled $1, 2, \ldots, n$, are rolled, and the sum is recorded. How should the dice's sides be weighted so that the resulting sum is closest to the uniform distribution on $2, 3, \ldots, 2n$? We answer this question by explicitly identifying the optimal pair of dice. This resolves a question raised by Gasarch and Kruskal in 1999 in a surprising way. We present additional results for the case of more than two possibly unfair $n$-sided dice and for the hypothetical case where the weights on each die are permitted to be negative, but must still sum to one.

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arXiv2023

Plane-filling curves of small degree over finite fields

A plane curve $C$ in $\mathbb{P}^2$ defined over $\mathbb{F}_q$ is called plane-filling if $C$ contains every $\mathbb{F}_q$-point of $\mathbb{P}^2$. Homma and Kim, building on the work of Tallini, proved that the minimum degree of a smooth plane-filling curve is $q+2$. We study smooth plane-filling curves of degree $q+3$ and higher.

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arXiv2023

Plane curves giving rise to blocking sets over finite fields

In recent years, many useful applications of the polynomial method have emerged in finite geometry. Indeed, algebraic curves, especially those defined by Rédei-type polynomials, are powerful in studying blocking sets. In this paper, we reverse the engine and study when blocking sets can arise from rational points on plane curves over finite fields. We show that irreducible curves of low degree cannot provide blocking sets and prove more refined results for cubic and quartic curves. On the other hand, using tools from number theory, we construct smooth plane curves defined over $\mathbb{F}_p$ of degree at most $4p^{3/4}+1$ whose points form blocking sets.

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arXiv2022

Linear families of smooth hypersurfaces over finitely generated fields

Let $K$ be a finitely generated field. We construct an $n$-dimensional linear system $\mathcal{L}$ of hypersurfaces of degree $d$ in $\mathbb{P}^n$ defined over $K$ such that each member of $\mathcal{L}$ defined over $K$ is smooth, under the hypothesis that the characteristic $p$ does not divide $\gcd(d, n+1)$ (in particular, there is no restriction when $K$ has characteristic $0$). Moreover, we exhibit a counterexample when $p$ divides $\gcd(d, n+1)$.

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arXiv2022

The EKR-module property of pseudo-Paley graphs of square order

We prove that a family of pseudo-Paley graphs of square order obtained from unions of cyclotomic classes satisfies the Erdős-Ko-Rado (EKR) module property, in a sense that the characteristic vector of each maximum clique is a linear combination of characteristic vectors of canonical cliques. This extends the EKR-module property of Paley graphs of square order and solves a problem proposed by Godsil and Meagher. Different from previous works, which heavily rely on tools from number theory, our approach is purely combinatorial in nature. The main strategy is to view these graphs as block graphs of orthogonal arrays, which is of independent interest.

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arXiv2022

Van Lint-MacWilliams' conjecture and maximum cliques in Cayley graphs over finite fields

A well-known conjecture due to van Lint and MacWilliams states that if $A$ is a subset of $\mathbb{F}_{q^2}$ such that $0,1 \in A$, $|A|=q$, and $a-b$ is a square for each $a,b \in A$, then $A$ must be the subfield $\mathbb{F}_q$. This conjecture is often phrased in terms of the maximum cliques in Paley graphs. It was first proved by Blokhuis and later extended by Sziklai to generalized Paley graphs. In this paper, we give a new proof of the conjecture and its variants, and show this Erdős-Ko-Rado property of Paley graphs extends to a larger family of Cayley graphs, which we call Peisert-type graphs, resolving conjectures by Mullin and Yip.

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arXiv2022

Bijective Cremona transformations of the plane

We study the birational self-maps of the projective plane over finite fields that induce permutations on the set of rational points. As a main result, we prove that no odd permutation arises over a non-prime finite field of characteristic two, which completes the investigation initiated by Cantat about which permutations can be realized this way. Main ingredients in our proof include the invariance of parity under groupoid conjugations by birational maps, and a list of generators for the group of such maps.

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arXiv2021

Transverse lines to surfaces over finite fields

We prove that if $S$ is a smooth reflexive surface in $\mathbb{P}^3$ defined over a finite field $\mathbb{F}_q$, then there exists an $\mathbb{F}_q$-line meeting $S$ transversely provided that $q\geq c\operatorname{deg}(S)$, where $c=\frac{3+\sqrt{17}}{4}\approx 1.7808$. Without the reflexivity hypothesis, we prove the existence of a transverse $\mathbb{F}_q$-line for $q\geq \operatorname{deg}(S)^2$.

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arXiv2020

On the proportion of transverse-free plane curves

We study the asymptotic proportion of smooth plane curves over a finite field $\mathbb{F}_q$ which are tangent to every line defined over $\mathbb{F}_q$. This partially answers a question raised by Charles Favre. Our techniques include applications of Poonen's Bertini theorem and Schrijver's theorem on perfect matchings in regular bipartite graphs. Our main theorem implies that a random smooth plane curve over $\mathbb{F}_q$ admits a transverse $\mathbb{F}_q$-line with very high probability.

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