added a demo file

demo.m2

@@ -0,0 +1,59 @@
+--naming conventions for jet variables
+restart
+needsPackage "Jets"
+R=QQ[a,x_1,y_(1,1)]
+jets(2,R)
+--basic jets functionalities
+restart
+needsPackage "Jets"
+--define a singular curve
+R=QQ[x,y]
+I=ideal(y^2-x^3)
+--compute jets
+elapsedTime jets(3,I)
+--jets are cached, lower orders too
+I.cache.jet.jetsMatrix
+elapsedTime jets(3,I)
+elapsedTime jets(2,I)
+--jets applies to polynomial rings
+jets(3,R)
+ring jets(3,I) === jets(3,R)
+--and quotient rings
+Q=R/I
+jets(2,Q)
+--parametrize the curve
+T=QQ[t]
+f=map(T,Q,{t^2,t^3})
+isWellDefined f
+--jets apply to ring mays
+j2f=jets(2,f)
+--there is a jet projection (algebraically an inclusion)
+p32Q=jetsProjection(3,2,Q)
+--it is a natural transformation
+j3f=jets(3,f)
+p32T=jetsProjection(3,2,T)
+p32T * j2f === j3f * p32Q
+--compute the principal component of the jets
+P=principalComponent(2,I)
+--it should have the same dimension as the jet scheme
+dim P == dim jets(2,I)
+--it should be a component of the jet scheme
+dec=primaryDecomposition jets(2,I)
+any(dec, p -> p == P)
+--compute jets of varieties
+X=Spec Q
+jets(2,X)
+jets(2,X) === Spec jets(2,Q)
+--jets need not be homogeneous
+Proj jets(2,Q)
+--use projective grading to get homogeneous ideals
+degrees R
+jets(2,I)
+isHomogeneous oo
+degrees jets(0,R,Projective=>true)
+degrees jets(1,R,Projective=>true)
+degrees jets(2,R,Projective=>true)
+jets(2,I,Projective=>true)
+isHomogeneous oo
+--compute projective jets of varieties
+jets(2,X,Projective=>true)

paperExamples.m2

@@ -52,4 +52,3 @@ radical JI2==JI2
 P_0== principalComponent(1,I2,Saturate=>false)
 L= apply({P_0,I2}, X -> hilbertSeries(X,Reduce=>true))
 numerator first L == (numerator last L)^2
-