On a conjecture on exponential Diophantine equations

We study the solutions of a Diophantine equation of the form $a^x+b^y=c^z$, where $a\equiv 2 \pmod 4$, $b\equiv 3 \pmod 4$ and $\gcd (a,b,c)=1$. The main result is that if there exists a solution $(x,y,z)=(2,2,r)$ with $r>1$ odd then this is the only solution in integers greater than 1, with the possible exception of finitely many values $(c,r)$. We also prove the uniqueness of such a solution if any of $a$, $b$, $c$ is a prime power. In a different vein, we obtain various inequalities that must be satisfied by the components of a putative second solution

An inequality about irreducible factors of integer polynomials

AbstractWe give a new upper bound for the height of an irreducible factor of an integer polynomial. This paper also contains several bounds for the case of polynomials with complex coefficients

Applications of the representation of finite fields by matrices

AbstractWe consider the matrix well-known representation of K[X]/(P), when P is monic irreducible polynomial, with coefficients in K. This representation enables us to give a fast algorithm to solve the equation xd=a in a finite field

On mechanical quantifier elimination for elementary algebra and geometry

We give solutions to two problems of elementary algebra and geometry: (1) find conditlons on real numbers p, q, and r; so that the polynomial function f(x) = x4 + px2 + q x+ r is nonnegative for all real x and (2) find conditions on real numbers a, b, and c so that the ellipse (x−c)2q2+y2b2−1=0 lies inside the unit circle y2 + x2 - 1 = O. Our solutions are obtained by following the basic outline of the method of quantifier elimination by cylindrical algebraic decomposition (Collins, 1975), but we have developed, and have been considerably aided by, modified vcrsions of certain of its steps. We have found three equally simple but not obviously equivalent solutions for the first problem, illustrating the difficulty of obtaining unique “simplest” solutions to quantifier eliminetion problems of elementary algebra and geometry