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# standardSols -- find the "standard solutions" for the oscillator system associated to a graph

- Usage:
    `standardSols(G,R)`
    `standardSols(G)`
- Inputs:
  - G, a graph, an undirected, connected graph $G$
  - R, a ring, created with oscRing
- Optional inputs:
  - Reduced => a Boolean value, default value false, if true, then the angles are reduced to the first angle being 0. That is, the number of oscillators will be one less than the number of vertices of $G$
- Outputs:
  - an ideal, a list of the "standard solutions" that you get where all the angles are the same. This is the same as a certain Segre variety.

## Description

The oscillator ideal associated to a graph constructed by oscQuadrics always contains the "standard solutions" as a minimal prime. These standard solutions are the Segre variety $\mathbb{P}^1\times \mathbb{P}^{n-1}$, where $n$ is the number of vertices of the graph. These are the solutions where all the angles are the same.

    i1 : G = graph({0,1,2,3}, {{0,1},{1,2},{2,3},{0,3}});

    i2 : R = oscRing(G);

    i3 : I = oscQuadrics(G, R);

    o3 : Ideal of R

    i4 : any(decompose I, P -> P == standardSols(G, R))

    o4 = true

## See also

- oscRing -- create a polynomial ring for a given graph or number of oscillators
- oscSystem -- the ideal of the reduced equilibrium points of a dynamical system of oscillators
- oscJacobian -- create the Jacobian for the oscillator system associated to a graph

## Ways to use standardSols:

- `standardSols(Graph)`
- `standardSols(Graph,Ring)`

## For the programmer

The object standardSols is a method function with options.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Oscillators/Documentation.m2:444:0.