The OSCAR project
The OSCAR project
We will start with talks about data structures on Tuesday. Wednesday to Friday are reserved for coding and short discussions when necessary.
Modules are some of the most fundamental mathematical structures for computer algebra, once we move past rings. Due to their importance, and due to the many different applications in different areas they suffer from an overload of requirements (and too many inconsistent bad habits in different areas). This tries to define a compromise that allows both sophisticated users and beginners to use (at least) some modules for computations.
I will show how to model the Boolean algebra of constructible sets in an affine/projective ambient variety starting X from the category of free modules over the (homogeneous) coordinate ring of X.
Writing a positive polynomial as a sum of squares (SOS) is an important problem in computational mathematics, with many applications in continuous and combinatorial optimization. A natural question is to which extent can an approximated sum of squares decomposition be rounded off to an exact rational decomposition, so a to provide a non-negativity certificate. In this talk we first prove that if a rational polynomial is the sum of two squares in an algebraic extension of odd degree of the rational numbers, then it can always be decomposed as a rational SOS. For the case of more than two polynomials we provide an explicit example of a rational polynomial that is the sum of three squares with coefficients in Q(alpha), alpha the cubic root of 2, that cannot be decomposed as a rational SOS. We will show computations in Maple and Singular to answer these questions and find the decompositions.
You can suggest coding sprint topics on the Wiki