Quantifier elimination: Optimal solution for two classical examples
Journal of Symbolic Computation
Well … it isn't quite that simple
ACM SIGSAM Bulletin
ISSAC '92 Papers from the international symposium on Symbolic and algebraic computation
Integration to obtain expressions valid on domains of maximum extent
ISSAC '93 Proceedings of the 1993 international symposium on Symbolic and algebraic computation
The evaluation of trigonometric integrals avoiding spurious discontinuities
ACM Transactions on Mathematical Software (TOMS)
Positivity conditions for quartic polynomials
SIAM Journal on Scientific Computing
Encyclopedia of Optimization
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We consider a monic polynomial of even degree with symbolic coefficients. We give a method for obtaining an expression in the coefficients (regarded as parameters) that is a lower bound on the value of the polynomial, or in other words a lower bound on the minimum of the polynomial. The main advantage of accepting a bound on the minimum, in contrast to an expression for the exact minimum, is that the algebraic form of the result can be kept relatively simple. Any exact result for a minimum will necessarily require parametric representations of algebraic numbers, whereas the bounds given here are much simpler. In principle, the method given here could be used to find the exact minimum, but only for low degree polynomials is this feasible; we illustrate this for a quartic polynomial. As an application, we compute rectifying transformations for integrals of trigonometric functions. The transformations require the construction of polynomials that are positive definite.