Certificate Gaussoids

The Certificate for valuated Gaussoids of rank 4 contains a certificate which proves Theorem Conjecture 8.4 of

[BDKS18] T. Boege, A. D’Alì, T. Kahle and B. Sturmfels: The Geometry of Gaussoids, Found. Comput. Math. (2018). arXiv:1710.07175

It works in the polynomial ring

\[\mathbb Q[p_{I}^{\pm 1}\mid I\subseteq [n]][a_{\{i,j\}|K}^{\pm 1}\mid i,j \in [n] \text{ distinct}, K\subseteq [n]\setminus\{i,j\}]\]

in which

  • \(p_I\) represents the principal minors of a symmetric \(n\times n\) matrix

  • \(a_{\{i,j\}|K}\) represents the almost-principal minors of a symmetric \(n\times n\) matrix.

The script contains a generating set \(F\) for the ideal \(T_4\) generated by the polynomial relations among these minors up to saturation as specified in Definition 4.1 of [BDKS18]. For a specific weight vector \(w_{\text{max}}\in\mathbb R^{\{p_I,a_{\{i,j\}|K}\}}\), the script shows

  • \(w_{\text{max}}\in\text{Trop}(F)\) by verifying that all initial forms \(\text{in}_{w_{\text{max}}}(f)\) for \(f\in F\) have at least 2 terms

  • \(w_{\text{max}}\notin\text{Trop}(T_4)\) by verifying that the initial ideal \(\text{in}_{w_{\text{max}}}(T_4)\) contains the monomial \(a_{23}a_{23|1}\).

                    SINGULAR                              /  Development
A Computer Algebra System for Polynomial Computations    /   version 4.1.1
by: W. Decker, G.-M. Greuel, G. Pfister, H. Schoenemann  \   Feb 2018
FB Mathematik der Universitaet, D-67653 Kaiserslautern    \
> LIB "tropicalBasis.lib"; // initializes necessary libraries and helper functions
> intvec wMin = 14,10,6,0,6,8,8,2,8,6,6,2,8,8,8,8,8,4,2,10,9,3,5,5,9,11,
.   1,5,7,5,5,5,7,7,1,5,8,6,4,4; // wMin is in min-convention
> intvec wMax = -wMin;           // Singular uses max-convention
> intvec allOnes = onesVector(size(wMax));
> ring r = 0,(p,p1,p12,p123,p1234,p124,p13,p134,p14,p2,p23,p234,p24,p3,p34,p4,
.   a12,a12_3,a12_34,a12_4,a13,a13_2,a13_24,a13_4,a14,a14_2,a14_23,a14_3,
.   a23,a23_1,a23_14,a23_4,a24,a24_1,a24_13,a24_3,a34,a34_1,a34_12,a34_2),
.   (a(allOnes),a(wMax),lp); // prepending allOnes makes no difference mathematically
                             // as the ideal is homogeneous,
                             // but it helps computationally
> ideal F =                  // Singular ideals are list of polynomials
.   a34_12*a13_24+p124*a14_23-a14_2*p1234,
    [...]                    // see uploaded script
.   -p1*p2+a12^2+p*p12;
> ideal inF = initial(F,wMax); // initial forms of the elements in F
                               // all are at least binomial, hence wMas lies in Trop(F)
> ideal I = groebner(F);
> ideal inI = initial(I,wMax); // initial forms of all elements in the Gr\"obner basis
                               // this is a Groebner basis of in_wMax(I)
> NF(a23*a23_1,inI);           // normal form is 0 hence a23*a23_1 lies in in_wMax(I)