Added affine Euler char to CharacteristicClasses

Added affine Euler characteristic method to CharacteristicClasses package.

Author: Martin-Helmer

M2/Macaulay2/packages/CharacteristicClasses.m2

@@ -2,14 +2,14 @@
 
 newPackage(
      	"CharacteristicClasses",
-	Version =>"2.0",
-    	Date => "October 24, 2015",
+	Version =>"2.1",
+    	Date => "June 24, 2025",
     	Authors => {{Name => "Martin Helmer", 
-		  Email => "martin.helmer@berkeley.edu", 
-		  HomePage => "https://math.berkeley.edu/~mhelmer/"},
+		  Email => "martin.helmer@swansea.ac.uk", 
+		  HomePage => "http://martin-helmer.com/"},
 	      {Name => "Christine Jost", 
 		  Email => "[email protected]"}},
-	Headline => "CSM and Segre classes and the Euler characteristic for subschemes of smooth complete toric varieties",
+    	Headline => "CSM classes, Segre classes and the Euler characteristic",
 	Keywords => {"Intersection Theory"},
     	DebuggingMode => false,
 	PackageImports => { "Elimination", "PrimaryDecomposition", "NormalToricVarieties"},
@@ -22,6 +22,7 @@ newPackage(
 	     "published article URI" => "https://msp.org/jsag/2015/7-1/p04.xhtml",
 	     "published article DOI" => "10.2140/jsag.2015.7.31",
 	     "published code URI" => "https://msp.org/jsag/2015/7-1/jsag-v7-n1-x04-CharacteristicClasses.m2",
+	     "repository code URI" => "https://github.com/Macaulay2/M2/blob/master/M2/Macaulay2/packages/CharacteristicClasses.m2",
 	     "release at publication" => "82375d8c668f3acf1d825b8ba991081769fba742",	    -- git commit number in hex
 	     "version at publication" => "1.1",
 	     "volume number" => "7",
@@ -59,9 +60,46 @@ export{"Segre",
    "ProjectiveDegree",
    "PnResidual",
    "bertini",
-   "bertiniCheck"
+   "bertiniCheck", 
+   "EulerAffine"
    }
 
+euleraffinehyp := (eqn) -> (
+    S1:=ring eqn;
+    n:=numgens(S1)-1;
+    enleqn:=(homogenize(eqn,(entries vars S1)_0_0)*(entries vars S1)_0_0);
+    cuteqn:=(substitute(enleqn,{(entries vars S1)_0_0=>0}));
+    if enleqn==0 then enleqn=0_S1;
+    if cuteqn==0 then cuteqn=0_S1;
+    enI:=ideal(enleqn);
+    cuI:=ideal(cuteqn);
+    tem1:=CSM(enI);
+    A:=ring tem1;
+    tem2:=CSM(A,cuI);
+    tem:=tem1-tem2;
+    return (tem_(A_0^n))+1;
+);
+
+EulerAffine=method(TypicalValue=>ZZ);
+EulerAffine (Ideal) := idea -> (
+    S:=ring idea;
+    if not (isPolynomialRing S) then error("Sorry, expecting an ideal defined over a polynomial ring"); 
+    kk:=coefficientRing S;
+    n:=numgens S;
+    z:=getSymbol "zzz";
+    S=kk[z_0..z_n];
+    schem:=(map(S,ring idea,{z_1..z_n})) idea;
+    r:=numgens schem;
+    gschem:=(entries gens schem)_0;
+    sset:=new MutableList from {0..r};
+    psum:=new MutableList from {0..r};
+    for s from 1 to r do(
+	sset#s=apply(subsets(gschem,s),eq->euleraffinehyp(product(eq)));
+	psum#s=sum(sset#s)
+	);
+    return sum(1..r,s->-(-1)^s*psum#s);
+);
+
 MultiProjCoordRing=method(TypicalValue=>Ring);
 MultiProjCoordRing (Symbol,List):=(x,l)->(
     kk:=ZZ/32749;
@@ -2474,6 +2512,36 @@ doc ///
                time CSM(U,CheckSmooth=>false)
 
 ///
+
+doc ///
+     Key
+     	  EulerAffine
+	  (EulerAffine,Ideal)
+     Headline
+     	  The Euler Characteristic of an affine variety. 
+     Usage
+     	  EulerAffine I
+     Inputs
+     	  I:Ideal
+	    an ideal in a polynomial ring over a field defining an affine variety.
+     Outputs
+     	  :RingElement
+	   the Euler characteristic 
+Description
+     	  Text
+	       This command computes the Euler characteristic of a complex affine variety.  
+	  Example
+	       kk=ZZ/32749;
+	       R=kk[x_1..x_3]
+	       I=ideal(x_1^2+x_2^2+x_3^2-1)
+	       time EulerAffine I
+	  Text
+	      Observe that the algorithm is a probabilistic algorithm and may give a wrong answer with a small but nonzero probability. Read more under 
+	       @TO "probabilistic algorithm"@.
+///
+
+
+
 --------------------------------------------------------
 -- Tests
 --------------------------------------------------------
@@ -2546,6 +2614,19 @@ TEST ///
     assert(csmH#{0}==Chern(A,ideal(I_0)));
     assert(seg==(V//(1+V)));    
 ///
+
+TEST ///
+-*
+   restart
+   needsPackage "CharacteristicClasses"
+   installPackage "CharacteristicClasses"
+*-
+         kk=ZZ/32749;
+	 R=kk[x_1..x_3];
+	 I=ideal(x_1^2+x_2^2+x_3^2-1);
+	 eu=EulerAffine I;
+	 assert(eu==2);
+///
 -------------------------------------------------------
 -- References
 ------------------------------------------------------