It was conjectured by Edoukou in 2008 that a non-degenerate Hermitian threefold in P 4 ( F q 2 ) has at most d ( q 5 + q 2 ) + q 3 + 1 points in common with a threefold of degree d defined over F q 2. He proved the conjecture for d = 2. In this ...

In algebraic geometry, it is important to provide effective parametrizations for families of curves, both in theory and in practice. In this paper, we present such an effective parametrization for the moduli of genus-5 curves that are neither ...

We obtain certain algebraic invariants relevant in studying codes on subgroups of weighted projective tori inside an n-dimensional weighted projective space. As application, we compute all the main parameters of generalized toric codes on these ...$\mathbb{}\mathrm{}\mathrm{}$

We design and analyze an algorithm for computing solutions with coefficients in a finite field F q of underdetermined systems defined over F q. The algorithm is based on reductions to zero-dimensional searches. The searches are performed on “...

In this paper, we will study Ciani curves in characteristic p ≥ 3, in particular their standard forms C : x 4 + y 4 + z 4 + r x 2 y 2 + s y 2 z 2 + t z 2 x 2 = 0. It is well-known that any Ciani curve is a non-hyperelliptic curve of ...

Let ${\mathbb{F}}_q$ be a finite field and $E_b:y^2=x^3+b$ be an ordinary (i.e., non-supersingular) elliptic curve (of j-invariant 0) such that $\sqrt{b}\in {\mathbb{F}}_q$ and $q\not\equiv 1(\mathrm{mod}27)$. For example, these conditions are fulfilled for the curve BLS12-381 ($b=4$). It is a de facto standard ...${\mathrm{}\mathrm{}}^{}{}_{}{\mathbb{}}_{}$${\mathbb{}}_{}$${}_{}{\mathbb{}}_{}$${\mathbb{}}_{}$

We compute the parameters of the linear codes that are associated with all projective embeddings of Grassmann varieties.

We present an example of a subfield $\mathcal{F}\subset \mathbb{R}$ and a matrix A whose conventional and nonnegative ranks equal five, but the nonnegative rank with respect to $\mathcal{F}$ equals six. In other words, A can be represented as a sum of five rank-one matrices with ...$\mathcal{}$

This paper shows that there exists a plane curve of degree q 3 + 1 with exactly two inner Galois points, of which the smooth model is the Hermitian curve of degree q + 1, where q is a power of the characteristic p > 0. Similar ...

A rational distance set in the plane is a point set which has the property that all pairwise distances between its points are rational. Erd?s and Ulam conjectured in 1945 that there is no dense rational distance set in the plane. In this paper we ...

We give a complete characterization of all Galois subfields of the generalized GiuliettiKorchmros function fields Cn/Fq2n for n5. Calculating the genera of the corresponding fixed fields, we find new additions to the list of known genera of maximal ...

We propose a systematic method to produce potentially good recursive towers over finite fields. The graph point of view, so as some magma and sage computations are used in this process. We also establish some theoretical functional criterion ensuring ...

Let q be a positive power of a prime number. For arbitrary positive integers h, n, m with n dividing m and arbitrary $$\gamma ,\alpha \in {\mathbb {F}}_{q^m}$$ , Fqm with $$\gamma \ne 0$$ 0 the number of $${\mathbb {F}}_{q^m}$$Fqm-rational points of the ...

Segre (Ann Mat Pura Appl 48:1---96, 1959 ) mentioned that the number $$N$$ N of points on a curve which splits into $$k$$ k distinct lines on the projective plane over a finite field of order $$q$$ q satisfies $$kq - \frac{k(k-3)}{2} \le N \le kq+1.$$ k q - k ( k - 3 ) 2 ≤ N ≤ k q + 1 . We see that the upper bound is satisfactory, but the lower one ...

Let A be an abelian variety over a finite field k. The k-isogeny class of A is uniquely determined by the Weil polynomial f"A. For a given prime number @__ __<>chark we give a classification of group schemes B[@__ __], where B runs through the isogeny ...

Let ${\mathbb F}_q$ be a finite field of characteristic 2, not containing ${\mathbb F}_4$ . Let k 2 be an even integer. We give a full classification of quadratic forms over ${\mathbb F}_{q^k}$ of codimension 2 provided that certain three coefficients are from ${\mathbb F}_4$ . We apply this to the classification of ...

For a plane curve over F"q of degree q+1, it is known by our previous work that the number of its F"q-rational points is at most q^2+1. In this paper, we determine the curves that attain this maximum, up to projective equivalence.

We manage an upper bound for the number of rational points of a Frobenius nonclassical plane curve over a finite field. Together with previous results, the modified Sziklai conjecture is settled affirmatively.