# isBirational -- whether a rational map is birational

## Synopsis

• Usage:
isBirational phi
• Inputs:
• phi,
• Optional inputs:
• BlowUpStrategy => ..., default value "Eliminate",
• Certify => ..., default value false, whether to ensure correctness of output
• Verbose => ..., default value true,
• Outputs:
• , whether phi is birational

## Description

The testing passes through the methods projectiveDegrees, degreeMap and isDominant.

 i1 : GF(331^2)[t_0..t_4] o1 = GF 109561[t ..t ] 0 4 o1 : PolynomialRing i2 : phi = rationalMap(minors(2,matrix{{t_0..t_3},{t_1..t_4}}),Dominant=>infinity) o2 = -- rational map -- source: Proj(GF 109561[t , t , t , t , t ]) 0 1 2 3 4 target: subvariety of Proj(GF 109561[x , x , x , x , x , x ]) defined by 0 1 2 3 4 5 { x x - x x + x x 2 3 1 4 0 5 } defining forms: { 2 - t + t t , 1 0 2 - t t + t t , 1 2 0 3 2 - t + t t , 2 1 3 - t t + t t , 1 3 0 4 - t t + t t , 2 3 1 4 2 - t + t t 3 2 4 } o2 : RationalMap (quadratic dominant rational map from PP^4 to hypersurface in PP^5) i3 : time isBirational phi -- used 0.0635391 seconds o3 = true i4 : time isBirational(phi,Certify=>true) Certify: output certified! -- used 0.134831 seconds o4 = true