4,751 research outputs found
Existence problems of primitive polynomials over finite fields
This thesis concerns existence of primitive polynomials over finite fields with one coefficient
arbitrarily prescribed. It completes the proof of a fundamental conjecture of
Hansen and Mullen (1992), which asserts that, with some explicable general exceptions,
there always exists a primitive polynomial of any degree n over any finite field with an
arbitrary coefficient prescribed. This has been proved whenever n is greater than or equal to 9 or n is less than or equal to 3, but was
unestablished for n = 4, 5, 6 and 8.
In this work, we efficiently prove the remaining cases of the conjecture in a selfcontained
way and with little computation; this is achieved by separately considering
the polynomials with second, third or fourth coefficient prescribed, and in each case developing
methods involving the use of character sums and sieving techniques. When the
characteristic of the field is 2 or 3, we also use p-adic analysis.
In addition to proving the previously unestablished cases of the conjecture, we also
offer shorter and self-contained proof of the conjecture when the first coefficient of the
polynomial is prescibed, and of some other cases where the proof involved a large amount
of computation. For degrees n = 6, 7 and 8 and selected values of m, we also prove the
existence of primitive polynomials with two coefficients prescribed (the constant term and
any other coefficient)
On the number of -free elements with prescribed trace
In this paper we derive a formula for the number of -free elements over a
finite field with prescribed trace, in particular trace zero, in
terms of Gaussian periods. As a consequence, we derive a simple explicit
formula for the number of primitive elements, in quartic extensions of Mersenne
prime fields, having absolute trace zero. We also give a simple formula in the
case when is prime. More generally, for a positive integer
whose prime factors divide and satisfy the so called semi-primitive
condition, we give an explicit formula for the number of -free elements with
arbitrary trace. In addition we show that if all the prime factors of
divide , then the number of primitive elements in , with
prescribed non-zero trace, is uniformly distributed. Finally we explore the
related number, , of elements in with
multiplicative order and having trace . Let such that , where is the largest factor of
with the same radical as that of . We show there exists an element in
of (large) order with trace if and only if and . Moreover we derive an explicit formula for the
number of elements in with the corresponding large order
and having absolute trace zero, where is a Mersenne
prime
Primitive polynomials with prescribed second coefficient
The Hansen-Mullen Primitivity Conjecture (HMPC) (1992) asserts that, with some (mostly obvious) exceptions, there exists a primitive polynomial of degree n over any finite fieldwith any coefficient arbitrarily prescribed. This has recently been provedwhenever n ≥ 9. It is also known to be truewhen n ≤ 3.We showthat there exists a primitive polynomial of any degree n ≥ 4 over any finite field with its second coefficient (i.e., that of xn−2) arbitrarily prescribed. In particular, this establishes the HMPC when n = 4. The lone exception is the absence of a primitive polynomial of the form x4 + a1x3 + x2 + a3x + 1 over the binary field. For n ≥ 6 we prove a stronger result, namely that the primitive polynomialmay also have its constant termprescribed. This implies further cases of the HMPC. When the field has even cardinality 2-adic analysis is required for the proofs
Computing GCRDs of Approximate Differential Polynomials
Differential (Ore) type polynomials with approximate polynomial coefficients
are introduced. These provide a useful representation of approximate
differential operators with a strong algebraic structure, which has been used
successfully in the exact, symbolic, setting. We then present an algorithm for
the approximate Greatest Common Right Divisor (GCRD) of two approximate
differential polynomials, which intuitively is the differential operator whose
solutions are those common to the two inputs operators. More formally, given
approximate differential polynomials and , we show how to find "nearby"
polynomials and which have a non-trivial GCRD.
Here "nearby" is under a suitably defined norm. The algorithm is a
generalization of the SVD-based method of Corless et al. (1995) for the
approximate GCD of regular polynomials. We work on an appropriately
"linearized" differential Sylvester matrix, to which we apply a block SVD. The
algorithm has been implemented in Maple and a demonstration of its robustness
is presented.Comment: To appear, Workshop on Symbolic-Numeric Computing (SNC'14) July 201
Irreducible polynomials over with three prescribed coefficients
For any positive integers and , we prove that the number
of monic irreducible polynomials of degree over in which
the coefficients of , and are prescribed has
period as a function of , after a suitable normalization. A similar
result holds over , with the period being . We also show
that this is a phenomena unique to characteristics and . The result is
strongly related to the supersingularity of certain curves associated with
cyclotomic function fields, and in particular it complements an
equidistribution result of Katz.Comment: Incorporated referee comments. Accepted for publication in Finite
Fields App
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