Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userMonogenic fields arising from trinomials
84
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We call a polynomial monogenic if a root $theta$ has the property that $mathbb{Z}[theta]$ is the full ring of integers in $mathbb{Q}(theta)$. Consider the two families of trinomials $x^n + ax + b$ and $x^n + cx^{n-1} + d$. For any $n>2$, we show that these families are monogenic infinitely often and give some positive densities in terms of the coefficients. When $n=5$ or 6 and when a certain factor of the discriminant is square-free, we use the Montes algorithm to establish necessary and sufficient conditions for monogeneity, illuminating more general criteria given by Jakhar, Khanduja, and Sangwan using other methods. Along the way we remark on the equivalence of certain aspects of the Montes algorithm and Dedekinds index criterion.
rate research
Read More
Riffaut (2019) conjectured that a singular modulus of degree $hge 3$ cannot be a root of a trinomial with rational coefficients. We show that this conjecture follows from the GRH, and obtain partial unconditional results.
In this short note we give an expression for some numbers $n$ such that the polynomial $x^{2p}-nx^p+1$ is reducible.
Using arithmetic jet spaces, we attach perfectoid spaces to smooth schemes and to $delta$-morphisms of smooth schemes. We also study perfectoid spaces attached to arithmetic differential equations defined by some of the remarkable $delta$-morphisms appearing in the theory such as the $delta$-characters of elliptic curves and the $delta$-period map on modular curves.
Consider polynomials over ${rm GF}(2)$. We describe efficient algorithms for finding trinomials with large irreducible (and possibly primitive) factors, and give examples of trinomials having a primitive factor of degree $r$ for all Mersenne exponents $r = pm 3 bmod 8$ in the range $5 < r < 10^7$, although there is no irreducible trinomial of degree $r$. We also give trinomials with a primitive factor of degree $r = 2^k$ for $3 le k le 12$. These trinomials enable efficient representations of the finite field ${rm GF}(2^r)$. We show how trinomials with large primitive factors can be used efficiently in applications where primitive trinomials would normally be used.
Let $f_0(z) = exp(z/(1-z))$, $f_1(z) = exp(1/(1-z))E_1(1/(1-z))$, where $E_1(x) = int_x^infty e^{-t}t^{-1}{,d}t$. Let $a_n = [z^n]f_0(z)$ and $b_n = [z^n]f_1(z)$ be the corresponding Maclaurin series coefficients. We show that $a_n$ and $b_n$ may be expressed in terms of confluent hypergeometric functions. We consider the asymptotic behaviour of the sequences $(a_n)$ and $(b_n)$ as $n to infty$, showing that they are closely related, and proving a conjecture of Bruno Salvy regarding $(b_n)$. Let $rho_n = a_n b_n$, so $sum rho_n z^n = (f_0,odot f_1)(z)$ is a Hadamard product. We obtain an asymptotic expansion $2n^{3/2}rho_n sim -sum d_k n^{-k}$ as $n to infty$, where the $d_kinmathbb Q$, $d_0=1$. We conjecture that $2^{6k}d_k in mathbb Z$. This has been verified for $k le 1000$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
