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ollieclarke8787 committed Jan 12, 2024
1 parent 387258e commit 03ee2c8
Showing 1 changed file with 100 additions and 45 deletions.
145 changes: 100 additions & 45 deletions Valuations/Valuations.m2
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Original file line number Diff line number Diff line change
Expand Up @@ -10,7 +10,7 @@ newPackage("Valuations",
{Name => "Shelby Cox", Email => "[email protected]"},
{Name => "Courtney George", Email => "[email protected]"},
{Name => "Oliver Clarke", Email => "[email protected]", HomePage => "oliverclarkemath.com"}},
DebuggingMode => false,
DebuggingMode => true,
HomePage => "https://github.com/Macaulay2/Workshop-2023-Minneapolis/tree/valuations",
Configuration => {},
--PackageExports => {"LocalRings", "SubalgebraBases", "InvariantRing", "Tropical", "gfanInterface", "Binomials"}
Expand All @@ -32,7 +32,12 @@ export{"valuation",
"getMExponent",
"domain",
"codomain",
-- "valM",
"valM",
"primeConesOfIdeal",
"coneToMatrix",
"getMaxIndependent",
"positivity",
"coneToValuation",
"OrderedQQn",
"OrderedQQVector",
"orderedQQn"
Expand Down Expand Up @@ -233,14 +238,33 @@ localRingValuation LocalRing := R -> (
--------------------------------------------------------------------------------
-------------------------------- example77 Valuation ---------------------------
--------------------------------------------------------------------------------
-*
-- internalTropicalVariety -- simple tropical variety computation
-- input:
-- I : Ideal: prime, homogeneous
--
-- output:
-- T : tropical polyhedral fan trop(I) without weights
--
internalTropicalVariety = method()
internalTropicalVariety Ideal := I -> (
if not I.cache#?"TropicalVariety" then (
startCone := gfanTropicalStartingCone I;
T := gfanTropicalTraverse startCone;
I.cache#"TropicalVariety" = T_0;
);
I.cache#"TropicalVariety"
)


primeConesOfIdeal = I -> (
F:=tropicalVariety(I, IsHomogeneous=>true,Prime=>true);
--F:=tropicalVariety(I, IsHomogeneous=>true,Prime=>true);
F := internalTropicalVariety I;
r:=rays(F);
c:=maxCones(F);
cns := for i in c list(r_i);
inCns := for c in cns list (flatten entries( c * transpose matrix{toList(numColumns(c) : 1)}));
L:= for i from 0 to #cns-1 list (J = gfanBuchberger(I, "w" => -1*(inCns#i));
L:= for i from 0 to #cns-1 list (
J := gfanBuchberger(I, "w" => -1*(inCns#i));
H := gfanInitialForms(J, -1*(inCns#i), "ideal" =>true);
K := H_1;
if binomialIsPrime(ideal(K)) then cns#i);
Expand Down Expand Up @@ -273,7 +297,7 @@ positivity = (f, matL) -> (
finalScaledMats := {};
matList := for i from 0 to #matL-1 list entries matL_i;
for i from 0 to #matList-1 do (
scaledRows = {};
scaledRows := {};
for j from 0 to #(matList_i)-1 do (
coeff := -1*floor(min apply(#(matList_i)_j, k -> (((matList_i)_j)_k)/(flatten entries l)_k));
scaledRows = append(scaledRows, (1/gcd(flatten entries (coeff*l + matrix{(matList_i)_j})))*(coeff*l + matrix{(matList_i)_j}));
Expand All @@ -286,15 +310,21 @@ positivity = (f, matL) -> (
)

-- TODO need to generalize!
coneToValuation = (coneRays, I) -> (
F := tropicalVariety(I, IsHomogeneous=>true,Prime=>true);
coneToValuation = (coneRays, I, S) -> (
--F := tropicalVariety(I, IsHomogeneous=>true,Prime=>true);
F := internalTropicalVariety I;
M := coneToMatrix(coneRays);
scaledM := (positivity(F, {M}))/(i -> sub(i, ZZ));
T := QQ[e_1, e_2, e_3, y, MonomialOrder=>{Weights=>((entries scaledM_0)_0), Weights=>((entries scaledM_0)_1)}];
scaledM := (positivity(F, {-M}))/(i -> sub(i, ZZ));
weightList := for row in entries scaledM_0 list Weights => row;
e := symbol e;
y := symbol y;
T := QQ[e_1, e_2, e_3, y, MonomialOrder => weightList];
--T := QQ[e_1, e_2, e_3, y, MonomialOrder=>{Weights=>((entries scaledM_0)_0), Weights=>((entries scaledM_0)_1)}];
val := leadTermValuation(T);
orderedM := orderedQQn(2, {Lex});
func := (f -> (
valf := val(sub(f, T));
m := map(T, S, gens T);
valf := val(m f);
if valf == infinity then infinity else (
(gens orderedM)*(scaledM_0)*(valf)
)
Expand All @@ -306,20 +336,29 @@ coneToValuation = (coneRays, I) -> (
-- TODO need to generalize!
-- construct the new valuation by taking min
valM = (T, valMTwiddle) -> (
valMfunc = (g) -> (
valMfunc := (g) -> (
x := symbol x;
e := symbol e;
y := symbol y;
R := QQ[x_1, x_2, x_3, e_1, e_2, e_3, y, MonomialOrder => Eliminate 3];
I := ideal{x_1 + x_2 + x_3 - e_1, x_1*x_2 + x_1*x_3 + x_2*x_3 - e_2, x_1*x_2*x_3 - e_3, (x_1 - x_2)*(x_1 - x_3)*(x_2 - x_3) - y};
f := e_1^2*e_2^2 - 4*e_2^3 - 4*e_3*e_1^3 + 18*e_1*e_2*e_3 - 27*e_3^2 - y^2;
--I := ideal{x_1 + x_2 + x_3 - e_1, x_1*x_2 + x_1*x_3 + x_2*x_3 - e_2, x_1*x_2*x_3 - e_3, (x_1 - x_2)*(x_1 - x_3)*(x_2 - x_3) - y};
--f := e_1^2*e_2^2 - 4*e_2^3 - 4*e_3*e_1^3 + 18*e_1*e_2*e_3 - 27*e_3^2 - y^2;
I := ideal{R_0 + R_1 + R_2 - R_3, R_0*R_1 + R_0*R_2 + R_1*R_2 - R_4, R_0*R_1*R_2 - R_5, (R_0 - R_1)*(R_0 - R_2)*(R_1 - R_2) - R_6};
f := R_3^2*R_4^2 - 4*R_4^3 - 4*R_5*R_3^3 + 18*R_3*R_4*R_5 - 27*R_5^2 - R_6^2;
S := valMTwiddle#"domain";
m := map(S, R, matrix{{0,0,0}} | matrix {gens S});
gTwiddle := m (sub(g, R) % I);
maxTwiddle := gTwiddle % ideal(sub(f, S));
use T; -- something above changes the user's ring (what could it be?) let's assume it was T
--gTwiddle := m (sub(g, R) % I);
--maxTwiddle := gTwiddle % ideal(sub(f, S));
TtoR := map(R, T, (gens R)_{0 .. numgens T -1});
gTwiddle := m ((TtoR g) % I);
RtoS := map(S, R, {0_S, 0_S, 0_S} | gens S);
maxTwiddle := gTwiddle % ideal(RtoS f);
--use T; -- something above changes the user's ring (what could it be?) let's assume it was T
valMTwiddle(maxTwiddle)
);
valuation(valMfunc, T, valMTwiddle#"codomain")
)
*-

--------------------------------------------------------------------------------
-------------------------------- Documentation ---------------------------------
--------------------------------------------------------------------------------
Expand Down Expand Up @@ -652,7 +691,7 @@ doc ///
orderedQQn
///

-*

doc ///
Key
"valM"
Expand All @@ -663,42 +702,51 @@ doc ///
Add description!
Example
R = QQ[x_1, x_2, x_3];
A = subring {
x_1 + x_2 + x_3,
x_1*x_2 + x_1*x_3 + x_2*x_3,
x_1*x_2*x_3,
(x_1 - x_2)*(x_1 - x_3)*(x_2 - x_3)
};
S = QQ[e_1, e_2, e_3, y];
presMap = map(R, S, gens A);
I = ker presMap
-- The primes cones of the tropical variety:
C = primeConesOfIdeal I
-- turn them into weights:
flatten (C/coneToMatrix/(i -> positivity(tropicalVariety I, {i})))
-- create weight valuations on the polynomial ring S
v0 = coneToValuation(C#0, I);
v1 = coneToValuation(C#1, I);
v2 = coneToValuation(C#2, I);
use S;
Text
The primes cones of the tropical variety:
Example
-- C = primeConesOfIdeal I -- This line takes too long for an example
C = {
matrix {{-3, 22}, {-6, -2}, {14, -3}, {-9, -3}},
matrix {{22, -3}, {-2, -6}, {-3, -9}, {-3, 14}},
matrix {{-11, -2}, {1, 19}, {13, -6}, {-10, -6}}
}
Text
Turn them into weights.

Note that @TT "internalTropicalVariety"@ is NOT exported

REMOVE THIS TEXT AND NEXT EXAMPLE BEFORE RELEASE!
Example
debug Valuations
flatten (C/coneToMatrix/(i -> positivity(internalTropicalVariety I, {i})))
Text
create weight valuations on the polynomial ring S
Example
v0 = coneToValuation(C#0, I, S);
v1 = coneToValuation(C#1, I, S);
v2 = coneToValuation(C#2, I, S);
use S;
v0(e_1^2 + e_2*e_3 - y^3) -- lead term from e_1^2
v1(e_1^2 + e_2*e_3 - y^3) -- lead term from y^3
v2(e_1^2 + e_2*e_3 - y^3) -- lead term from e_2*e_3
-- create the induced valuation on the subring A
Text
create the induced valuation on the subring A
Example
vA0 = valM(R, v0);
vA1 = valM(R, v1);
vA2 = valM(R, v2);
use R;
vA0(x_1^2 + x_2^2 + x_3^2)
vA1(x_1^2 + x_2^2 + x_3^2)
vA2(x_1^2 + x_2^2 + x_3^2)
Expand All @@ -708,18 +756,18 @@ doc ///
vA2((x_1^2 - x_2^2)*(x_1^2 - x_3^2)*(x_2^2 - x_3^2))

vA0(0_R)
-- Note, for elements not in A, the valuation returns nonsense
-- because the valuation does not come from a weight valuation
-- on R
Text
Note, for elements not in A, the valuation returns nonsense
because the valuation does not come from a weight valuation
on R
Example
vA0(x_2)
vA0(x_2^2)
vA0(x_2^3)
vA0(x_2^3)
SeeAlso

///

*-

doc ///
Key
Expand Down Expand Up @@ -954,3 +1002,10 @@ testPackage = x -> (
installPackage("Valuations");
check Valuations;
)

---------------------
restart
uninstallPackage "Valuations"
restart
installPackage "Valuations"
needsPackage "Valuations"

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