Effectivity Issues and Results for Hilbert's 17th Problem

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Gives constructive, effective versions of Artin's solution to Hilbert 17th with explicit degree and bit-size bounds for representing a nonnegative polynomial as sums of squares of rational functions, using quantifier elimination via cylindrical algebraic decomposition and elimination-of-variables strategies. Provides algorithms that output Positivstellensatz certificates and counts complexity: bounds are huge (tower/exponential in number of variables and input degree) so certificates are provably computable but often impractically large. Includes concrete examples and implemented decision steps.

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" chapters (15)

1. 0:00Start
2. 0:47Positivity and sums of squares
3. 6:36Outline of Artin's proof
4. 8:58Remaining problems
5. 18:04Degree of an incompatibility
6. 21:15Positivstellensatz: proofs
7. 22:37Positivstellensatz: Constructive proofs
8. 24:40Positivstellensatz implies Hilbert 17th problem
9. 27:16Weak inferences: first example
10. 33:24Weak inferences: case by case reasoning
11. 36:30List of statements needed into weak inferences form
12. 38:08List of statements into weak inferences form
13. 39:03Why a tower of five exponentials ?
14. 41:06What can be hoped for ??
15. 43:15What can be hoped for??