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lines changed Original file line number Diff line number Diff line change @@ -9,10 +9,10 @@ There is a linear relation `Σ = A(S)`.
99The linear relation reads: `p` belongs to `Σ` iff there exists `q` in `S` such that `A(q) = p`. 
1010This allows defining a variable bridge that would create variables `p` and substitute `A(q)` for `p` but this is not the purpose of this bridge. 
1111This bridge exploit the following alternative read: 
12- `p` belongs to `Σ` iff there exists `q` in `S` such that ``q in A^{-1}(p)`` where `A^{-1}` is the preimage of `p`. 
13- This preimage can be obtained as `A^\\ dagger p + \\ mathrm{ker}(A)` where `A^\\ dagger` is the pseudo-inverse of `A`. 
12+ `p` belongs to `Σ` iff there exists `q` in `S` such that ``q \\ in A^{-1}(p)`` where `` A^{-1}` ` is the preimage of `p`. 
13+ This preimage can be obtained as `` A^\\ dagger p + \\ mathrm{ker}(A)``  where `` A^\\ dagger` ` is the pseudo-inverse of `A`. 
1414It turns out that for polynomial bases indexed by monomials, `A` is close to row echelon form so 
15- `A^\\ dagger` and `\\ mathrm{ker}(A)` can easily be obtained. 
15+ `` A^\\ dagger``  and `` \\ mathrm{ker}(A)` ` can easily be obtained. 
1616
1717This is best described in an example. 
1818Consider the SOS constraint for the polynomial `p = 2x^4 + 2x^3 * y - x^2 * y^2 + 5y^4` 
 
 
   
 
     
   
   
          
    
    
     
    
      
     
     
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