Complexity and real computation
Complexity and real computation
On reducing a system of equations to a single equation
ISSAC '04 Proceedings of the 2004 international symposium on Symbolic and algebraic computation
The complexity of quantifier elimination and cylindrical algebraic decomposition
Proceedings of the 2007 international symposium on Symbolic and algebraic computation
Randomized NP-completeness for p-adic rational roots of sparse polynomials in one variable
Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation
Interpolation of functions related to the integer factoring problem
WCC'05 Proceedings of the 2005 international conference on Coding and Cryptography
Faster p-adic feasibility for certain multivariate sparse polynomials
Journal of Symbolic Computation
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Consider any nonzero univariate polynomial with rational coefficients, presented as an elementary algebraic expression (using only integer exponents). Letting 驴(f) denotes the additive complexity of f, we show that the number of rational roots of f is no more than 15 + 驴(f)2(24.01)驴(f)驴(f)!. This provides a sharper arithmetic analogue of earlier results of Dima Grigoriev and Jean-Jacques Risler, which gave a bound of C驴(f)2 for the number of real roots of f, for 驴(f) sufficiently large and some constant C with 1Cp-adic rationals, roots of bounded degree over a number field, and geometrically isolated roots of multivariate polynomial systems. We thus extend earlier bounds of Hendrik W. Lenstra, Jr. and the author to encodings more efficient than monomial expansions. We also mention a connection to complexity theory and note that our bounds hold for a broader class of fields.