Varieties over finite fields Algorithms Polynomials Rational points Complexity Computational complexity Problem solving Theorem proving Algebra Geometric solutions Homotopy algorithms Mathematical models Matrix algebra Polynomial system solving Probabilistic algorithms Boundary value problems Conditioning Effective elimination theory Effective first Bertini theorem Estimation Finite fields First Bertini theorem Geometry Lower complexity Normal complete intersection Partial differential equations Pebble game; time-space tradeoff; straight-line program; elimination theory Polynomial equation solving Second Bertini theorem Semi linear parabolic problems