AGA problem answers

### How to edit?

- Read a documentation page for wiki math markup or go to the page of any problem and click
**source**to see how other people write.

### Status of problems

**Problems with solutions**appear in bold.*Problems with partial solutions, hints, discussion*are in italics.- The other posted problems have a wiki page with the statement of the problem only.

Many problems have no wiki page at all… If you solved a problem (completely or partially), have an idea or a hint, or simply want to promote the discussion by posting the problem statement to the wiki, then **add the problem and create its page** and see your `wikidot` karma improve.

### Macaulay2 exercises

### Exercises by day

**Mon, June 18**:- Convex Algebraic Geometry:
**Newton Polytopes and Sums of Squares**,**Motzkin Form**, Sums of Squares on a Lemniscate, Convex Hulls of Planar Curves,**Conical Hull of a Compact Convex Set**,**Nonnegative and SOS cones** - Tropical Exercises:
*Minkowski sums*, Non-constant coefficients. - Computational algebraic geometry: Every system = a system of quadrics,
*Gamma exceptional set*, …

- Convex Algebraic Geometry:
**Tue, June 19**:- Computational algebraic geometry:
*Study coordinates*, Grassmannian of lines - Convexity:
**Polar convex body**, … - Groebner bases:
**Ascending Chain Condition**,**Elimination Example**, Lex Groebner basis and weight orders, …

- Computational algebraic geometry:
**Wed, June 20**:- Lifts of convex sets:
**Lemniscate of Gerono**

- Lifts of convex sets:
**Thu, June 21**:- Lifts of Convex Sets:
**Cross Polytopes**,**Rank of a Slack Matrix**,**Vanishing Ideals of Finitely Many Points**,**Birkhoff Polytope and Permutahedron** - Groebner Fans and Tropical Varieties:
**Principal Ideals**,*Term Orders and Weights*, Tropical Varieties and Gfan - Semidefinite and Polynomial Optimization:
**Explicit Example of Semidefinite Duality**, Univariate Nonnegative Polynomial

- Lifts of Convex Sets:
**Fri, June 22**:- Computational algebraic geometry:
**Descartes's Circle Theorem via elimination**,**Intersection of Ideals and Radical Membership** - Semidefinite and Polynomial Optimization:
**Stability number of a graph**,**Positive maps and decomposable maps**

- Computational algebraic geometry:
**Mon, June 25**:- Mixed Volume and Bernstein's Theorem:
*Newton polygons with no common slopes*,**Mixed volume computation**,**MV(P,P,...,P)**, Mixed volume is symmetric, Mixed Volume is multi-linear, Mixed volume and containment,**Bezout Bound**, Mixed volume formula in dimension 2, Criterion for non-zero mixed volume - Convex Algebraic Geometry:
**Dual Faces**, Maximal Faces of the Nonnegative Cone,**Exposed Extreme Rays of the Nonnegative Cone** - Primary Decomposition:
**Intersection of primary ideals**

- Mixed Volume and Bernstein's Theorem:
**Tues, June 26**:- Primary Decompositions:
**Splitting polynomial and saturations**,*Saturation by Elimination* - Computational Algebraic Geometry:
**Binomial start system**,**Deflation**. - Binomial Ideals: Cellular binomial ideals, Using binomial ideals for combinatorics

- Primary Decompositions:
**Thu, June 28**:- Computational Algebraic Geometry: Numerical primary decomposition, Macaulay dual space.
- Tropical:
**Projection of tropical hypersurface**, Minkowski sum and mixed faces

**Fri, June 29**:- Convex Algebraic Geometry:
*Inner product on the vector space of polynomials*,**Extreme rays of a spectrahedron**, Extreme rays of the cone of sums of squares - Primary Decomposition:
**Radicals of zero-dimensional ideals** - Tropical:
**Dimension Formula**,**Finite Combinatorial Types**, Dim 3 Resultant Polytopes

- Convex Algebraic Geometry:
**Mon, July 02**:- Biochemical Reaction Networks: 1-site phosphorylation,
- (Convex) Algebraic Geometry: Olympic Rings,
**Product of the unit disk**,*Associahedron*,**Cycle space of K5****Note:**exercise 1 (planar curves) is on Greg's first exercise sheet, see Mon, June 18

**Tues, July 03**:- Tropical Geometry:
**Valuations**,**Residue field**, p-adic GrÃ¶bner fans

- Tropical Geometry: