find the generators of the ideal defining zariski closure for kimura 3 parameter model

The Zariski closure of a set of points in an algebraic variety is the smallest algebraic variety that contains the set of points. For the Kimura 3-parameter model in algebraic geometry and algebraic statistics, we focus on the ideal generated by the relations defining this model.

### Kimura 3-Parameter Model

The Kimura 3-parameter (K3P) model is used in phylogenetics and is a model for nucleotide substitution. It allows for three different parameters to describe the mutation rates among the nucleotides. This model is associated with the group \( G = \mathbb{Z}_2 \times \mathbb{Z}_2 \) acting on a 4-dimensional vector space.

Given this model, the Zariski closure can be described by an ideal in the polynomial ring \( \mathbb{C}[p_{ij}] \), where \( p_{ij} \) represents the probability of a transition from state \( i \) to state \( j \). The ideal is generated by the quadratic relations that express the symmetries and constraints of the model.

### Finding the Generators of the Ideal

For the Kimura 3-parameter model, the generators of the ideal can be found by considering the following types of relations:

1. **Trace relations**: These relate to the fact that the matrix of transition probabilities is stochastic, meaning that the sum of each row is 1.
2. **Symmetry relations**: These come from the symmetries in the model, reflecting that certain transitions should have the same probability due to the model’s constraints.
3. **Determinant relations**: These are quadratic relations derived from the structure of the transition matrix, which is constrained by the group structure \( G = \mathbb{Z}_2 \times \mathbb{Z}_2 \).

For the K3P model, specifically, the generators of the ideal \( I \) in \( \mathbb{C}[p_{ij}] \) can be described as:

\[
I = \langle p_{ij}p_{kl} - p_{il}p_{kj} \text{ for } i, j, k, l \in \{1, 2, 3, 4\} \rangle
\]

These generators express the condition that for any 2x2 submatrix of the 4x4 transition matrix, the determinant should be zero. This comes from the fact that under the K3P model, the determinant of the transition matrix minus identity should vanish because the eigenvalues of the transition matrix are constrained by the group action.

### Summary of Generators

In summary, the generators of the ideal defining the Zariski closure for the Kimura 3-parameter model include:
- Quadratic polynomials expressing symmetry constraints.
- Relations derived from the stochasticity of the transition matrix.
- Determinant conditions on 2x2 submatrices of the transition matrix.

These generators define the Zariski closure of the variety corresponding to the Kimura 3-parameter model in the space of transition matrices.

Would you like to dive deeper into any part of this explanation, or do you have any questions? 

Here are 8 questions related to this topic:

1. What are the roles of the parameters in the Kimura 3-parameter model?
2. How does the group \( \mathbb{Z}_2 \times \mathbb{Z}_2 \) influence the ideal generators?
3. What is the significance of trace relations in the context of the K3P model?
4. How can we explicitly compute the determinant relations for the Kimura model?
5. What are the implications of these ideal generators on phylogenetic tree reconstruction?
6. Can these ideal generators be extended to other substitution models?
7. What algebraic tools are used to verify the correctness of these generators?
8. How does the Zariski topology influence the understanding of model constraints?

**Tip:** In algebraic statistics, understanding the generators of an ideal can give insight into the geometric structure of a statistical model, helping to identify the fundamental constraints of the model.

Explore the generators of the ideal defining the Zariski closure for the Kimura 3-parameter model in algebraic geometry and statistics. Understand the quadratic relations and constraints derived from the model's symmetries and stochastic properties.

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