Trace of totally positive algebraic integers and integer transfinite diameter
نویسنده
چکیده
Explicit auxiliary functions can be used in the “Schur-SiegelSmyth trace problem”. In the previous works, these functions were constructed only with polynomials having all their roots positive. Here, we use several polynomials with complex roots, which are found with Wu’s algorithm, and we improve the known lower bounds for the absolute trace of totally positive algebraic integers. This improvement has a consequence for the search of Salem numbers that have a negative trace. The same method also gives a small improvement of the upper bound for the integer transfinite diameter of [0,1].
منابع مشابه
The integer transfinite diameter of intervals and totally real algebraic integers
In this paper we build on some recent work of Amoroso, and Borwein and Erdélyi to derive upper and lower estimates for the integer transfinite diameter of small intervals + 03B4], is a fixed rational and 03B4 ~ 0. We also study functions g-, g, g+ associated with transfinite diameters of Farey intervals. Then we consider certain polynomials, which we call critical polynomials, associated to a g...
متن کاملThe integer Chebyshev problem
We are concerned with the problem of minimizing the supremum norm on an interval of a nonzero polynomial of degree at most n with integer coefficients. This is an old and hard problem that cannot be exactly solved in any nontrivial cases. We examine the case of the interval [0, 1] in most detail. Here we improve the known bounds a small but interesting amount. This allows us to garner further i...
متن کاملSalem Numbers of Trace -2 and Traces of Totally Positive Algebraic Integers
Until recently, no Salem numbers were known of trace below −1. In this paper we provide several examples of trace −2, including an explicit infinite family. We establish that the minimal degree for a Salem number of trace −2 is 20, and exhibit all Salem numbers of degree 20 and trace −2. Indeed there are just two examples. We also settle the closely-related question of the minimal degree d of a...
متن کاملAlgebraic integers whose conjugates all lie in an ellipse
We find all 15909 algebraic integers whose conjugates all lie in an ellipse with two of them nonreal, while the others lie in the real interval [−1, 2]. This problem has applications to finding certain subgroups of SL(2,C). We use explicit auxiliary functions related to the generalized integer transfinite diameter of compact subsets of C. This gives good bounds for the coefficients of the minim...
متن کاملComputing totally positive algebraic integers of small trace
We construct minimal polynomials of totally positive algebraic integers of small absolute trace by consideration of their reductions modulo auxiliary polynomials. Many new examples of such polynomials of minimal absolute trace (for given degree) are found. The computations are pushed to degrees that previously were unattainable, and one consequence is that the new examples form the majority of ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- Math. Comput.
دوره 78 شماره
صفحات -
تاریخ انتشار 2009