Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn the enumeration of polynomials with prescribed factorization pattern
59
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We use generating functions over group rings to count polynomials over finite fields with the first few coefficients prescribed and a factorization pattern prescribed. In particular, we obtain different exact formulas for the number of monic $n$-smooth polynomial of degree $m$ over a finite field, as well as the number of monic $n$-smooth polynomial of degree $m$ with the prescribed trace coefficient.
rate research
Read More
We obtain estimates on the number $|mathcal{A}_{boldsymbol{lambda}}|$ of elements on a linear family $mathcal{A}$ of monic polynomials of $mathbb{F}_q[T]$ of degree $n$ having factorization pattern $boldsymbol{lambda}:=1^{lambda_1}2^{lambda_2}cdots n^{lambda_n}$. We show that $|mathcal{A}_{boldsymbol{lambda}}|= mathcal{T}(boldsymbol{lambda}),q^{n-m}+mathcal{O}(q^{n-m-{1}/{2}})$, where $mathcal{T}(boldsymbol{lambda})$ is the proportion of elements of the symmetric group of $n$ elements with cycle pattern $boldsymbol{lambda}$ and $m$ is the codimension of $mathcal{A}$. Furthermore, if the family $mathcal{A}$ under consideration is sparse, then $|mathcal{A}_{boldsymbol{lambda}}|= mathcal{T}(boldsymbol{lambda}),q^{n-m}+mathcal{O}(q^{n-m-{1}})$. Our estimates hold for fields $mathbb{F}_q$ of characteristic greater than 2. We provide explicit upper bounds for the constants underlying the $mathcal{O}$--notation in terms of $boldsymbol{lambda}$ and $mathcal{A}$ with good behavior. Our approach reduces the question to estimate the number of $mathbb{F}_q$--rational points of certain families of complete intersections defined over $mathbb{F}_q$. Such complete intersections are defined by polynomials which are invariant under the action of the symmetric group of permutations of the coordinates. This allows us to obtain critical information concerning their singular locus, from which precise estimates on their number of $mathbb{F}_q$--rational points are established.
Let $D$ be a negative integer congruent to $0$ or $1bmod{4}$ and $mathcal{O}=mathcal{O}_D$ be the corresponding order of $ K=mathbb{Q}(sqrt{D})$. The Hilbert class polynomial $H_D(x)$ is the minimal polynomial of the $j$-invariant $ j_D=j(mathbb{C}/mathcal{O})$ of $mathcal{O}$ over $K$. Let $n_D=(mathcal{O}_{mathbb{Q}( j_D)}:mathbb{Z}[ j_D])$ denote the index of $mathbb{Z}[ j_D]$ in the ring of integers of $mathbb{Q}(j_D)$. Suppose $p$ is any prime. We completely determine the factorization of $H_D(x)$ in $mathbb{F}_p[x]$ if either $p mid n_D$ or $p mid D$ is inert in $K$ and the $p$-adic valuation $v_p(n_D)leq 3$. As an application, we analyze the key space of Oriented Supersingular Isogeny Diffie-Hellman (OSIDH) protocol proposed by Col`o and Kohel in 2019 which is the roots set of the Hilbert class polynomial in $mathbb{F}_{p^2}$.
Given an infinite set of special divisors satisfying a mild regularity condition, we prove the existence of a Borcherds product of non-zero weight whose divisor is supported on these special divisors. We also show that every meromorphic Borcherds product is the quotient of two holomorphic ones. The proofs of both results rely on the properties of vector valued Eisenstein series for the Weil representation.
We study the distribution of extensions of a number field $k$ with fixed abelian Galois group $G$, from which a given finite set of elements of $k$ are norms. In particular, we show the existence of such extensions. Along the way, we show that the Hasse norm principle holds for $100%$ of $G$-extensions of $k$, when ordered by conductor. The appendix contains an alternative purely geometric proof of our existence result.
We give an algorithm that finds a sequence of approximations with Dirichlet coefficients bounded by a constant only depending on the dimension. The algorithm uses the LLL-algorithm for lattice basis reduction. We present a version of the algorithm that runs in polynomial time of the input.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
