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Deterministically Testing Sparse Polynomial Identities of Unbounded Degree
, 2008
"... We present two deterministic algorithms for the arithmetic circuit identity testing problem. The running time of our algorithms is polynomially bounded in s and m, where s is the size of the circuit and m is an upper bound on the number monomials with nonzero coefficients in its standard representa ..."
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We present two deterministic algorithms for the arithmetic circuit identity testing problem. The running time of our algorithms is polynomially bounded in s and m, where s is the size of the circuit and m is an upper bound on the number monomials with nonzero coefficients in its standard representation. The running time of our algorithms also has a logarithmic dependence on the degree of the polynomial but, since a circuit of size s can only compute polynomials of degree at most 2 s, the running time does not depend on its degree. Before this work, all such deterministic algorithms had a polynomial dependence on the degree and therefore an exponential dependence on s. Our first algorithm works over the integers and it requires only blackbox access to the given circuit. Though this algorithm is quite simple, the analysis of why it works relies on Linnik’s Theorem, a deep result from number theory about the size of the smallest prime in an arithmetic progression. Our second algorithm, unlike the first, uses elementary arguments and works over any integral domains, but it uses the circuit in a less restricted manner. In both cases the running time has a logarithmic dependence on the largest coefficient of the polynomial.
ON THE GREATEST PRIME FACTOR OF p − 1 WITH EFFECTIVE CONSTANTS
"... Abstract. Let p denote a prime. In this article we provide the first published lower bounds for the greatest prime factor of p − 1 exceeding (p − 1) 1 2 in which the constants are effectively computable. As a result we prove that it is possible to calculate a value x0 such that for every x>x0 there ..."
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Abstract. Let p denote a prime. In this article we provide the first published lower bounds for the greatest prime factor of p − 1 exceeding (p − 1) 1 2 in which the constants are effectively computable. As a result we prove that it is possible to calculate a value x0 such that for every x>x0 there is a p<xwith the greatest prime factor of p − 1 exceeding x 3 5. The novelty of our approach is the avoidance of any appeal to Siegel’s Theorem on primes in arithmetic progression. 1.
Sieve Methods
"... Sieve methods have had a long and fruitful history. The sieve of Eratosthenes (around 3rd century B.C.) was a device to generate prime numbers. Later Legendre used it in his studies of the prime number counting function π(x). Sieve methods bloomed and became a topic of intense investigation after th ..."
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Sieve methods have had a long and fruitful history. The sieve of Eratosthenes (around 3rd century B.C.) was a device to generate prime numbers. Later Legendre used it in his studies of the prime number counting function π(x). Sieve methods bloomed and became a topic of intense investigation after the pioneering work of Viggo Brun (see [Bru16],[Bru19], [Bru22]). Using his formulation of the sieve Brun proved, that the sum
IN SEARCH OF SMALL EXPONENTIAL SUMS
"... Résumé. Soit f(x) une fraction rationnelle à coefficients entiers, vérifiant des hypothèses assez générales. On prouve l’existence d’une infinité d’entiers n, ayant exactement deux facteurs premiers, tels que la somme d’exponentielles ∑n x=1 exp ( 2πif(x)/n) soit en O(n 1 2 −βf), où βf> 0 est une co ..."
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Résumé. Soit f(x) une fraction rationnelle à coefficients entiers, vérifiant des hypothèses assez générales. On prouve l’existence d’une infinité d’entiers n, ayant exactement deux facteurs premiers, tels que la somme d’exponentielles ∑n x=1 exp ( 2πif(x)/n) soit en O(n 1 2 −βf), où βf> 0 est une constante ne dépendant que de la géométrie de f. On donne aussi des résultats de répartition du type Sato–Tate, pour certaines sommes de Salié, modulo n, avec n entier comme ci–dessus.