# projectionToHypersurface -- Generic projection to a hypersurface

## Synopsis

• Usage:
projectionToHypersurface I
projectionToHypersurface R
• Inputs:
• I, an ideal, an ideal in a polynomial ring
• R, a ring, a quotient of a polynomial ring
• Optional inputs:
• Codimension => an integer, default value null, specified if you already know the codimension of your Ideal (or QuotientRing) in your ambient ring
• MaxCoordinatesToReplace => an integer, default value infinity, to be passed to randomCoordinateChange
• Replacement => , default value Binomial, to be passed to randomCoordinateChange
• Homogeneous (missing documentation) => , default value true, to be passed to randomCoordinateChange
• Verbose => , default value false, set to true for verbose output
• Outputs:
• , a list with two entries, the generic projection map, and the ideal if I was provided, or the ring if R was provided

## Description

This creates a projection to a hypersurface. It differs from genericProjection(codim I - 1, I) as it only tries to find a hypersurface equation that vanishes along the projection, instead of finding one that vanishes exactly at the projection. This can be faster, and can be useful for finding points.

 i1 : R=ZZ/5[x,y,z]; i2 : I = ideal(random(3,R)-2, random(2,R)); o2 : Ideal of R i3 : projectionToHypersurface(I) ZZ 6 5 4 2 5 6 o3 = (map(R,--[y..z],{x, y - 2z}), ideal(y - 2y z - 2y z - 2y*z + z - 5 ------------------------------------------------------------------------ 2 2 2y z + y*z - 1)) o3 : Sequence i4 : projectionToHypersurface(R/I) R o4 = (map(------------------------------------------------------------------- 3 2 3 2 2 2 2 3 (- 2x - 2x*y - 2y + x z - y z + 2x*z - 2y*z - 2z - 2, - 2x*z ------------------------------------------------------------------------ ZZ --[y..z] 5 -------------,---------------------------------------------------------- 2 6 5 4 2 2 4 5 3 2 2 3 + 2y*z - 2z ) y - y z + 2y z - 2y z - y*z + 2y - y z - y*z + z + ------------------------------------------------------------------------ ZZ --[y..z] 5 -,{y + 2z, x + z}), ---------------------------------------------------- 6 5 4 2 2 4 5 3 2 2 1 y - y z + 2y z - 2y z - y*z + 2y - y z - y*z + ------------------------------------------------------------------------ -------) 3 z + 1 o4 : Sequence

If you already know the codimension is c, you can set Codimension=>c so the function does not compute it.

• genericProjection -- finds a random (somewhat) generic projection of the ring or ideal

## Ways to use projectionToHypersurface :

• "projectionToHypersurface(Ideal)"
• "projectionToHypersurface(Ring)"

## For the programmer

The object projectionToHypersurface is .