Summaries of recent relevant papers on Algebraic Geometry from arXiv

Summary of Recent Relevant Papers on Algebraic Geometry from arXiv

# Algebraic Geometry

## Short Description of Research Question
Algebraic Geometry is a branch of mathematics that studies the solutions of systems of polynomial equations using abstract algebraic techniques. It has close connections to other areas such as number theory, complex geometry, and mathematical physics. The recent literature reflects advances in algebraic geometry theory, algorithmic and computational methods, and applications including machine learning and dynamical systems.

## Summary of Work

1. **Equiresidual Algebraic Geometry I: The Affine Theory** by Jean Barbet (2019)
   This work generalizes classical algebraic geometry to fields that are not algebraically closed, developing foundations based on normic forms and an equiresidual version of the Nullstellensatz. It introduces new classes of algebras and radicals that lead to a dualization of affine algebraic varieties in these more general fields, connecting to model-theoretic algebraic geometry and scheme theory.

2. **Machine Learning Algebraic Geometry for Physics** by Jiakang Bao et al. (2022)
   This paper reviews applications of machine learning techniques to problems in algebraic geometry and physics. It discusses how algebraic geometry problems can be reframed as tensor mappings suitable for supervised and unsupervised learning, highlighting AI's role in advancing mathematical understanding.

3. **Algorithms in Real Algebraic Geometry: A Survey** by Saugata Basu (2014)
   This survey covers algorithmic developments related to real algebraic geometry, focusing on quantifier elimination, computation of topological invariants, and complexity. It bridges decision problems and computational hardness with recent numerical approaches.

4. **Test Ideals via Algebras of p^-e-Linear Maps** by Manuel Blickle (2009)
   The paper simplifies the treatment of test ideals by viewing them as minimal objects in certain modules over algebras of p^{-e}-linear operators, providing an elementary approach that extends known results and re-proves discreteness of jumping numbers.

5. **On the Algebraic Geometry of Polynomial Dynamical Systems** by Abdul S. Jarrah and Reinhard Laubenbacher (2008)
   Focuses on polynomial dynamical systems over finite fields, demonstrating how algebraic geometry facilitates solving problems related to structure, dynamics, and control theory in such systems.

6. **Numerical Algebraic Geometry for Macaulay2** by Anton Leykin (2009)
   Introduces numerical algebraic geometry methods using numerical polynomial homotopy continuation, blending symbolic and numerical computations to efficiently describe algebraic varieties.

7. **Multiplier Ideals in Algebraic Geometry** by Samuel Grushevsky (2005)
   An expository text on multiplier ideals discussing vanishing theorems, bounds on multiplicities in divisors, asymptotic constructions, and analytic interpretations in algebraic geometry.

8. **Amoebas of Algebraic Varieties and Tropical Geometry** by Grigory Mikhalkin (2004)
   A survey on amoebas (images via logarithmic maps) and their degeneration to tropical varieties, linking to tropical algebraic geometry with applications to tropical curves and beyond.

9. **Polyhedral Methods in Numerical Algebraic Geometry** by Jan Verschelde (2008)
   Discusses certificates for algebraic curves via Puiseux series and the computation of tropisms linked to mixed volumes, useful for preprocessing in numerical algebraic geometry.

10. **Birational Geometry of Cluster Algebras** by Mark Gross et al. (2013)
    Provides a geometric interpretation of cluster varieties through blowups of toric varieties, proving the Laurent phenomenon and addressing conjectures about cluster algebras.

## Papers
- Equiresidual algebraic geometry I: The affine theory (arXiv:1912.00347v3)
- Machine Learning Algebraic Geometry for Physics (arXiv:2204.10334v1)
- Algorithms in Real Algebraic Geometry: A Survey (arXiv:1409.1534v1)
- Test ideals via algebras of p^{-e}-linear maps (arXiv:0912.2255v3)
- On the algebraic geometry of polynomial dynamical systems (arXiv:0803.1825v1)
- Numerical Algebraic Geometry for Macaulay2 (arXiv:0911.1783v2)
- Multiplier ideals in algebraic geometry (arXiv:math/0502387v3)
- Amoebas of algebraic varieties and tropical geometry (arXiv:math/0403015v1)
- Polyhedral Methods in Numerical Algebraic Geometry (arXiv:0810.2983v1)
- Birational geometry of cluster algebras (arXiv:1309.2573v2)