Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userBounds on Factors in Z[x]
101
0
0.0
(
0
)
Authors
John Abbott
Ask ChatGPT about the research
No Arabic abstract
We gather together several bounds on the sizes of coefficients which can appear in factors of polynomials in Z[x]; we include a new bound which was latent in a paper by Mignotte, and a few minor improvements to some existing bounds. We compare these bounds and show that none is universally better than the others. In the second part of the paper we give several concrete examples of factorizations where the factors have unexpectedly large coefficients. These examples help us understand why the bounds must be larger than you might expect, and greatly extend the collection published by Collins.
rate research
Read More
We give a sufficient criterion for a lower bound of the cactus rank of a tensor. Then we refine that criterion in order to be able to give an explicit sufficient condition for a non-redundant decomposition of a tensor to be minimal and unique.
In this note we show how the irrationality measure of $zeta(s) = pi^2/6$ can be used to obtain explicit lower bounds for $pi(x)$. We analyze the key ingredients of the proof of the finiteness of the irrationality measure, and show how to obtain good lower bounds for $pi(x)$ from these arguments as well. Whi
The paper contained a preliminary version of a general theory of reciprocity laws on vector spaces.
We show that for all large enough $x$ the interval $[x,x+x^{1/2}log^{1.39}x]$ contains numbers with a prime factor $p > x^{18/19}.$ Our work builds on the previous works of Heath-Brown and Jia (1998) and Jia and Liu (2000) concerning the same problem for the longer intervals $[x,x+x^{1/2+epsilon}].$ We also incorporate some ideas from Harmans book `Prime-detecting sieves (2007). The main new ingredient that we use is the iterative argument of Matomaki and Radziwi{l}{l}(2016) for bounding Dirichlet polynomial mean values, which is applied to obtain Type II information. This allows us to take shorter intervals than in the above-mentioned previous works. We have also had to develop ideas to avoid losing any powers of $log x$ when applying Harmans sieve method.
In this paper, we study factorizations in the additive monoids of positive algebraic valuations $mathbb{N}_0[alpha]$ of the semiring of polynomials $mathbb{N}_0[X]$ using a methodology introduced by D. D. Anderson, D. F. Anderson, and M. Zafrullah in 1990. A cancellative commutative monoid is atomic if every non-invertible element factors into irreducibles. We begin by determining when $mathbb{N}_0[alpha]$ is atomic, and we give an explicit description of its set of irreducibles. An atomic monoid is a finite factorization monoid (FFM) if every element has only finitely many factorizations (up to order and associates), and it is a bounded factorization monoid (BFM) if for every element there is a bound for the number of irreducibles (counting repetitions) in each of its factorizations. We show that, for the monoid $mathbb{N}_0[alpha]$, the property of being a BFM and the property of being an FFM are equivalent to the ascending chain condition on principal ideals (ACCP). Finally, we give various characterizations for $mathbb{N}_0[alpha]$ to be a unique factorization monoid (UFM), two of them in terms of the minimal polynomial of $alpha$. The properties of being finitely generated, half-factorial, and other-half-factorial are also investigated along the way.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
