O #Math 676 (Comp. Alg. Geom.) Lab 2 (Spring 2009)
#Handy note:
current one.
[control]-k adds a new Maple input line above the
O with(Groebner);
O # Here's how you find leading terms w.r.t. various term orders
O LeadingTerm(x^3*y^2*z + x^2*y^3*z^2 +x^2*y^4*z, plex(x,y,z));
#regular old lex (pure lex)
O LeadingTerm(x^3*y^2*z + x^2*y^3*z^2 +x^2*y^4*z, grlex(x,y,z));
#graded lex
O LeadingTerm(x^3*y^2*z + x^2*y^3*z^2 +x^2*y^4*z, tdeg(x,y,z));
#grevlex (total degree, for some reason)
O sort(x^3*y^2*z + x^2*y^3*z^2 +x^2*y^4*z, [x,y,z], tdeg);
#grevlex again, sorting this time
O f := 3*x^2*y - 8*y^3 + 2*x*y^2 - 10*x^2 - y; #Here's another
polynomial
O InitialForm(f, grlex(x,y)); #We can grab all terms with maximal
degree
O #You can also mess around with weighted orders. Instead of
considering just the sum of the exponents as the degree, we can
take a weighted sum:
O LeadingTerm(x^4*y^2*z + x^2*y^4*z, wdeg([1/2,2,1],[x,y,z]));
#This makes y much more important (and x less).
InitialForm(x^4*y^2*z + x^2*y^4*z, wdeg([1/2,2,1],[x,y,z]));
#Here's the whole initial form - just one mon.
O LeadingTerm(x^4*y^2*z + x^2*y^4*z, wdeg([2,1,1],[x,y,z]));
#This makes x much more important than y,z.
InitialForm(x^4*y^2*z + x^2*y^4*z, wdeg([2,1,1],[x,y,z])); #Now
the initial form is just the other mon.
O LeadingTerm(x^4*y^2*z + x^2*y^4*z, wdeg([1,1,1],[x,y,z]));
#graded lex in a funny form.
InitialForm(x^4*y^2*z + x^2*y^4*z, wdeg([1,1,1],[x,y,z]));
#Here's the whole initial form - two monomials!
#This is a boundary order, i.e., the leading
#monomial changes with wdegs just above or below.
O # 1. Cook up a polynomial that gives a different leading term
for each of the three standard monomial orders.
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# Now for Groebner bases!
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F2 := [x^3-2*x*y, x^2*y-2*y^2+x];
Basis(F2, plex(x,y,z));
#Hey! Something nontrivial! Let's try a different order:
G := Basis(F2, grlex(x,y,z)); #Here's graded lex.
Here's one from class:
F1 := [x-z,y-z];
Basis(F1, plex(x,y,z));
# OK, so that shows that our generators do give a Groebner
basis. Let's try something harder:
O #So, at least in Maple, different orders led to different
Groebner bases.
#We didn't get to this last time, but Maple is actually giving a
reduced basis, which is unique (for a fixed
#term order). Thus, a GB in one order won't be one in another:
O Basis(G, plex(x,y,z));
O #2. Can you cook up an example (maybe in three variables) that
has a different GB for plex, grlex, and tdeg? (NOTE: I haven't
tried this, but I see no reason why there wouldn't be one....)
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O
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#So how do we divide?
NormalForm(x^2*y + x*y^2 + y^2, [y^2-1, x*y-1], plex(x,y));
NormalForm(x^2*y + x*y^2 + y^2, [x*y-1, y^2-1], plex(x,y));
# Right, it's that mean example from class.
O #3. Is the polynomial above actually in the ideal generated by
the divisors??
O # Let's see how fast we can go on something a little bigger.
O # We'll wipe out the memory of the session, just in case there's
something useful in there....
restart: with(Groebner): F := [a*b*c+d*e*f, a^2*b^2-c*d+e^2*f,
a*b*c*d*e-f^4, a^2-b^2+c^2*d]:
Basis(F, plex(a,b,c,d,e,f));
O restart: with(Groebner): F := [a*b*c+d*e*f, a^2*b^2-c*d+e^2*f,
a*b*c*d*e-f^4, a^2-b^2+c^2*d]:
Basis(F, tdeg(a,b,c,d,e,f));
O #Notice anything about how fast the above two lines took? Same
ideal in each case - only the term order changed.... Strangely,
plex usually provides a larger basis -- this is just a special
case, I suppse.
O #4. Can you cook up a gnarly example that takes Maple more than
a few seconds? What if you try a different monomial order?
(HINT: Try higher degrees and/or more variables/polynomials
and/or nastier (rational) coefficients and/or coefficients with
a bunch of different primes.)
O #5. Write a short routine that chooses a random set of
polynomials (in 3 variables, say) and checks whether the given
basis is actually a GB. If you run this 100 (or 1000...) times,
how often is your random choice a GB? For that matter, how can
we check whether a given basis is indeed Groebner? (HINT: Maybe
encode the S-criterion?) (NOTE: I don't know the answer for
this one, but it's worth playing around.....)
O #6. Try out some of the above operations over different fields,
just for kicks.
O #You could also check out the PolynomialIdeals package in the
help pages -- in there, you can define an ideal, and it will
automatically remember various characteristics like any Groebner
bases that you have computed....
O #7. You could check out EliminationIdeal(). We will talk about
elimination shortly, but the point is that from a lex GB, you
can extract (hopefully) an equation in one variable, one in two,
one in three, etc., so that a univariate solver can be employed
for the univariate equation. See the next example.
O with(PolynomialIdeals):
TC := PolynomialIdeal([y-x^2, z-x^3]); #Here's an ideal for the
twisted cubic.
O BTC := Basis(TC, plex(x,y,z)); #Here's a lex GB for it.
O IsZeroDimensional(TC); #Maple will tell you whether the variety
is zero-dimensional.
O EliminationIdeal(TC,{x}); #Here's how you intersect ideal TC
with k[x] to get polynomial(s) in one variable, if there are
any. Why aren't there? Now try {y} and {z}??
O EliminationIdeal(TC,{x,y}); #Here we intersect with {x,y}.
This works better....
O # 8. Now try the previous with a zero-dimensional ideal. You
can cook one up or try something from the notes or just use [3*
x^2-4*y*z, x*z-y^3, x^2+z] (randomly chosen). Use a tree of
elimination ideals ({x}, {x,y}, {x,y,z} and solve() (or similar)
to "solve" the system.
O #Here's a fun ideal:
IllEx := PolynomialIdeal([(x^2+y^2+z^2-1)*(y-x^2)*(x-1/2), (x^2+
y^2+z^2-1)*(z-x^3)*(y-1/2), (x^2+y^2+z^2-1)*(y-x^2)*(z-x^3)*
(z-1/2)], characteristic=0);
O EqDecomp := EquidimensionalDecomposition(IllEx); #This cooks up
a set of ideals, each with different dimension:
O EqDecomp[1]; #Here's the 0-dim part.
HilbertDimension(EqDecomp[1]);
O EqDecomp[2]; #Dim 1.
HilbertDimension(EqDecomp[2]);
O EqDecomp[3]; #Dim 2.
HilbertDimension(EqDecomp[3]);
O # 9. Can we actually get to the isolated solutions?
SED1 := Solve(EqDecomp[1]);
O # 10. What does all that mean? If you want, mess around with
the Groebner and PolynomialIdeals packages and see if you can
find the actual solution(s) in dimension zero....
O # 11. Another approach would be to try some small system of
linears for which you know the answer....