Elimination Theory in the 21st century Carlos D'Andrea NSF-CBMS Conference on Applications of Polynomial Systems Carlos D'Andrea Elimination Theory in the 21st century Session on Open Problems Friday 3.30 pm BYOP 5 minutes per presentation to book a time slot, contact Hal or me Before we start... Carlos D'Andrea Elimination Theory in the 21st century BYOP 5 minutes per presentation to book a time slot, contact Hal or me Before we start... Session on Open Problems Friday 3.30 pm Carlos D'Andrea Elimination Theory in the 21st century 5 minutes per presentation to book a time slot, contact Hal or me Before we start... Session on Open Problems Friday 3.30 pm BYOP Carlos D'Andrea Elimination Theory in the 21st century to book a time slot, contact Hal or me Before we start... Session on Open Problems Friday 3.30 pm BYOP 5 minutes per presentation Carlos D'Andrea Elimination Theory in the 21st century Before we start... Session on Open Problems Friday 3.30 pm BYOP 5 minutes per presentation to book a time slot, contact Hal or me Carlos D'Andrea Elimination Theory in the 21st century Apart from being crazy about applications and implementations... The Paradigm of the 21st Century Carlos D'Andrea Elimination Theory in the 21st century and implementations... The Paradigm of the 21st Century Apart from being crazy about applications Carlos D'Andrea Elimination Theory in the 21st century The Paradigm of the 21st Century Apart from being crazy about applications and implementations... Carlos D'Andrea Elimination Theory in the 21st century The Paradigm of the 21st Century Apart from being crazy about applications and implementations... Carlos D'Andrea Elimination Theory in the 21st century The number8 and \size" of the solutions of > F1(X1;:::; Xn) = 0 > < F2(X1;:::; Xn) = 0 . > . > : Fn(X1;:::; Xn) = 0 where \size" of Fi = (d; L) is bounded by and generically equal to d n; nd n−1L \Cost" of Elimination Carlos D'Andrea Elimination Theory in the 21st century where \size" of Fi = (d; L) is bounded by and generically equal to d n; nd n−1L \Cost" of Elimination The number8 and \size" of the solutions of > F1(X1;:::; Xn) = 0 > < F2(X1;:::; Xn) = 0 . > . > : Fn(X1;:::; Xn) = 0 Carlos D'Andrea Elimination Theory in the 21st century is bounded by and generically equal to d n; nd n−1L \Cost" of Elimination The number8 and \size" of the solutions of > F1(X1;:::; Xn) = 0 > < F2(X1;:::; Xn) = 0 . > . > : Fn(X1;:::; Xn) = 0 where \size" of Fi = (d; L) Carlos D'Andrea Elimination Theory in the 21st century \Cost" of Elimination The number8 and \size" of the solutions of > F1(X1;:::; Xn) = 0 > < F2(X1;:::; Xn) = 0 . > . > : Fn(X1;:::; Xn) = 0 where \size" of Fi = (d; L) is bounded by and generically equal to d n; nd n−1L Carlos D'Andrea Elimination Theory in the 21st century Carlos D'Andrea Elimination Theory in the 21st century Moreover: the complexity of computing Gr¨obnerbases is double exponential in( d; L) The output is already exponential!!! Carlos D'Andrea Elimination Theory in the 21st century the complexity of computing Gr¨obnerbases is double exponential in( d; L) The output is already exponential!!! Moreover: Carlos D'Andrea Elimination Theory in the 21st century The output is already exponential!!! Moreover: the complexity of computing Gr¨obnerbases is double exponential in( d; L) Carlos D'Andrea Elimination Theory in the 21st century \Relax" the input (and expect the output to be \relaxed" too!) Change the Computational Model What can we do then??? Carlos D'Andrea Elimination Theory in the 21st century (and expect the output to be \relaxed" too!) Change the Computational Model What can we do then??? \Relax" the input Carlos D'Andrea Elimination Theory in the 21st century Change the Computational Model What can we do then??? \Relax" the input (and expect the output to be \relaxed" too!) Carlos D'Andrea Elimination Theory in the 21st century What can we do then??? \Relax" the input (and expect the output to be \relaxed" too!) Change the Computational Model Carlos D'Andrea Elimination Theory in the 21st century Tropical Geometry \Relaxed" resultants Implicitization matrices (Wednesday) Syzygies and Rees Algebras (Wednesday) ::: Relaxing the input Carlos D'Andrea Elimination Theory in the 21st century \Relaxed" resultants Implicitization matrices (Wednesday) Syzygies and Rees Algebras (Wednesday) ::: Relaxing the input Tropical Geometry Carlos D'Andrea Elimination Theory in the 21st century Implicitization matrices (Wednesday) Syzygies and Rees Algebras (Wednesday) ::: Relaxing the input Tropical Geometry \Relaxed" resultants Carlos D'Andrea Elimination Theory in the 21st century (Wednesday) Syzygies and Rees Algebras (Wednesday) ::: Relaxing the input Tropical Geometry \Relaxed" resultants Implicitization matrices Carlos D'Andrea Elimination Theory in the 21st century Syzygies and Rees Algebras (Wednesday) ::: Relaxing the input Tropical Geometry \Relaxed" resultants Implicitization matrices (Wednesday) Carlos D'Andrea Elimination Theory in the 21st century (Wednesday) ::: Relaxing the input Tropical Geometry \Relaxed" resultants Implicitization matrices (Wednesday) Syzygies and Rees Algebras Carlos D'Andrea Elimination Theory in the 21st century ::: Relaxing the input Tropical Geometry \Relaxed" resultants Implicitization matrices (Wednesday) Syzygies and Rees Algebras (Wednesday) Carlos D'Andrea Elimination Theory in the 21st century Relaxing the input Tropical Geometry \Relaxed" resultants Implicitization matrices (Wednesday) Syzygies and Rees Algebras (Wednesday) ::: Carlos D'Andrea Elimination Theory in the 21st century Probabilistic algorithms Computations \over the Reals" \Parallelization" Homotopy methods (Tomorrow) ::: Changing the model Carlos D'Andrea Elimination Theory in the 21st century Computations \over the Reals" \Parallelization" Homotopy methods (Tomorrow) ::: Changing the model Probabilistic algorithms Carlos D'Andrea Elimination Theory in the 21st century \Parallelization" Homotopy methods (Tomorrow) ::: Changing the model Probabilistic algorithms Computations \over the Reals" Carlos D'Andrea Elimination Theory in the 21st century Homotopy methods (Tomorrow) ::: Changing the model Probabilistic algorithms Computations \over the Reals" \Parallelization" Carlos D'Andrea Elimination Theory in the 21st century (Tomorrow) ::: Changing the model Probabilistic algorithms Computations \over the Reals" \Parallelization" Homotopy methods Carlos D'Andrea Elimination Theory in the 21st century ::: Changing the model Probabilistic algorithms Computations \over the Reals" \Parallelization" Homotopy methods (Tomorrow) Carlos D'Andrea Elimination Theory in the 21st century Changing the model Probabilistic algorithms Computations \over the Reals" \Parallelization" Homotopy methods (Tomorrow) ::: Carlos D'Andrea Elimination Theory in the 21st century Disclaimer These are not the highlights of the 21st century (so far), just a sampling of results All these areas are very active and promising right now, more will appear in the future... A lot of important names and interesting areas will be omitted...sorry! Some \Tapas" into the 21st century Carlos D'Andrea Elimination Theory in the 21st century Disclaimer These are not the highlights of the 21st century (so far), just a sampling of results All these areas are very active and promising right now, more will appear in the future... A lot of important names and interesting areas will be omitted...sorry! Some \Tapas" into the 21st century Carlos D'Andrea Elimination Theory in the 21st century These are not the highlights of the 21st century (so far), just a sampling of results All these areas are very active and promising right now, more will appear in the future... A lot of important names and interesting areas will be omitted...sorry! Some \Tapas" into the 21st century Disclaimer Carlos D'Andrea Elimination Theory in the 21st century All these areas are very active and promising right now, more will appear in the future... A lot of important names and interesting areas will be omitted...sorry! Some \Tapas" into the 21st century Disclaimer These are not the highlights of the 21st century (so far), just a sampling of results Carlos D'Andrea Elimination Theory in the 21st century , more will appear in the future... A lot of important names and interesting areas will be omitted...sorry! Some \Tapas" into the 21st century Disclaimer These are not the highlights of the 21st century (so far), just a sampling of results All these areas are very active and promising right now Carlos D'Andrea Elimination Theory in the 21st century A lot of important names and interesting areas will be omitted...sorry! Some \Tapas" into the 21st century Disclaimer These are not the highlights of the 21st century (so far), just a sampling of results All these areas are very active and promising right now, more will appear in the future... Carlos D'Andrea Elimination Theory in the 21st century sorry! Some \Tapas" into the 21st century Disclaimer These are not the highlights of the 21st century (so far), just a sampling of results All these areas are very active and promising right now, more will appear in the future... A lot of important names and interesting areas will be omitted... Carlos D'Andrea Elimination Theory in the 21st century Some \Tapas" into the 21st century Disclaimer These are not the highlights of the 21st century (so far), just a sampling of results All these areas are very active and promising right now, more will appear in the future... A lot of important names and interesting areas will be omitted...sorry!