regularInCodimension -- attempts to show that the ring is regular in codimension n

Usage:

regularInCodimension(n, R)
Inputs:
- n, an integer,
- R, a ring, an equidimensional quotient of a polynomial ring over a field
Optional inputs:
- Modulus => a number, default value 0, work modulo the given prime modulus
- PairLimit => a number, default value 100, passed to isCodimAtLeast
- SPairsFunction => a function, default value FunctionClosure[../FastMinors.m2:174:23-174:42], passed to isCodimAtLeast
- UseOnlyFastCodim => a Boolean value, default value false, tell the function not to use the built in dim command and only use isCodimAtLeast
- MinMinorsFunction => a function, default value FunctionClosure[../FastMinors.m2:169:26-169:40], control how many minors are computed before computing codim
- MaxMinors => a function, default value FunctionClosure[../FastMinors.m2:164:18-164:50], how many minors to consider before giving up
- CodimCheckFunction => a function, default value FunctionClosure[../FastMinors.m2:170:27-170:46], control how many minors to compute in between calls to codim
- DetStrategy => ..., default value Cofactor, DetStrategy is a strategy for allowing the user to choose how determinants (or rank), is computed
- PointOptions => ..., default value {Strategy => Default, Homogeneous => false, Replacement => Binomial, ExtendField => true, PointCheckAttempts => 0, DecompositionStrategy => Decompose, NumThreadsToUse => 1, DimensionFunction => dim, Verbose => false}, options to pass to functions in the package RandomPoints
- Strategy => ..., default value new OptionTable from {Points => 0, Random => 16, GRevLexLargest => 0, LexSmallestTerm => 16, LexLargest => 0, LexSmallest => 16, GRevLexSmallestTerm => 16, RandomNonzero => 16, GRevLexSmallest => 16}, strategies for choosing submatrices
- Verbose => ..., default value false
- VerifyNonRegular => ..., default value false
Outputs:
- true, if the ring is regular in codimension n, false if it determines it is not, and null if no determination is made

Description

This function returns true if R is regular in codimension n, false if it is not, and null if it did not make a determination. We assume that R is equidimensional and it is a quotient ring of a polynoimal ring over a field. It considers interesting minors of the jacobian matrix to try to verify that the ring is regular in codimension n. It is frequently much faster at giving an affirmative answer than computing the dimension of the ideal of all minors of the Jacobian. We begin with a simple example which is R1, but not R2.

i1 : R = QQ[x, y, z]/ideal(x*y-z^2);

i2 : regularInCodimension(1, R)

o2 = true

i3 : regularInCodimension(2, R)

o3 = false

Next we consider a more interesting example that is R1 but not R2, and highlight the speed differences. Note that regularInCodimension(2, R) returns nothing, as the function did not determine whether the ring was regular in codimension n.

i4 : T = ZZ/101[x1,x2,x3,x4,x5,x6,x7];

i5 : I =  ideal(x5*x6-x4*x7,x1*x6-x2*x7,x5^2-x1*x7,x4*x5-x2*x7,x4^2-x2*x6,x1*x4-x2*x5,x2*x3^3*x5+3*x2*x3^2*x7+8*x2^2*x5+3*x3*x4*x7-8*x4*x7+x6*x7,x1*x3^3*x5+3*x1*x3^2*x7+8*x1*x2*x5+3*x3*x5*x7-8*x5*x7+x7^2,x2*x3^3*x4+3*x2*x3^2*x6+8*x2^2*x4+3*x3*x4*x6-8*x4*x6+x6^2,x2^2*x3^3+3*x2*x3^2*x4+8*x2^3+3*x2*x3*x6-8*x2*x6+x4*x6,x1*x2*x3^3+3*x2*x3^2*x5+8*x1*x2^2+3*x2*x3*x7-8*x2*x7+x4*x7,x1^2*x3^3+3*x1*x3^2*x5+8*x1^2*x2+3*x1*x3*x7-8*x1*x7+x5*x7);

o5 : Ideal of T

i6 : S = T/I;

i7 : dim S

o7 = 3

i8 : time regularInCodimension(1, S)
 -- used 0.977593s (cpu); 0.747284s (thread); 0s (gc)

o8 = true

i9 : time regularInCodimension(2, S)
 -- used 9.02218s (cpu); 6.81142s (thread); 0s (gc)

There are numerous examples where regularInCodimension is several orders of magnitude faster that calls of dim singularLocus.

The following is a (pruned) affine chart on an Abelian surface obtained as a product of two elliptic curves. It is nonsingular, as our function verifies. If one does not prune it, then the dim singularLocus call takes an enormous amount of time, otherwise the running times of dim singularLocus and our function are frequently about the same.

i10 : R = QQ[c, f, g, h]/ideal(g^3+h^3+1,f*g^3+f*h^3+f,c*g^3+c*h^3+c,f^2*g^3+f^2*h^3+f^2,c*f*g^3+c*f*h^3+c*f,c^2*g^3+c^2*h^3+c^2,f^3*g^3+f^3*h^3+f^3,c*f^2*g^3+c*f^2*h^3+c*f^2,c^2*f*g^3+c^2*f*h^3+c^2*f,c^3-f^2-c,c^3*h-f^2*h-c*h,c^3*g-f^2*g-c*g,c^3*h^2-f^2*h^2-c*h^2,c^3*g*h-f^2*g*h-c*g*h,c^3*g^2-f^2*g^2-c*g^2,c^3*h^3-f^2*h^3-c*h^3,c^3*g*h^2-f^2*g*h^2-c*g*h^2,c^3*g^2*h-f^2*g^2*h-c*g^2*h,c^3*g^3+f^2*h^3+c*h^3+f^2+c);

i11 : dim(R)

o11 = 2

i12 : time (dim singularLocus (R))
 -- used 0.0260008s (cpu); 0.0265814s (thread); 0s (gc)

o12 = -1

i13 : time regularInCodimension(2, R)
 -- used 0.574316s (cpu); 0.397534s (thread); 0s (gc)

o13 = true

i14 : time regularInCodimension(2, R)
 -- used 0.402835s (cpu); 0.286795s (thread); 0s (gc)

o14 = true

i15 : time regularInCodimension(2, R)
 -- used 0.304325s (cpu); 0.217823s (thread); 0s (gc)

o15 = true

The function works by choosing interesting looking submatrices, computing their determinants, and periodically (based on a logarithmic growth setting), computing the dimension of a subideal of the Jacobian. The option Verbose can be used to see this in action.

i16 : time regularInCodimension(2, S, Verbose=>true)
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing LexSmallest
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing LexSmallest
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing LexSmallestTerm
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing LexSmallestTerm
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing LexSmallestTerm
 -- internalChooseMinor: Choosing LexSmallest
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing LexSmallest
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing LexSmallest
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing LexSmallest
 -- internalChooseMinor: Choosing LexSmallestTerm
 -- internalChooseMinor: Choosing LexSmallestTerm
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing LexSmallestTerm
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing LexSmallestTerm
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing LexSmallest
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing LexSmallest
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing LexSmallestTerm
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing LexSmallest
 -- internalChooseMinor: Choosing LexSmallestTerm
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing LexSmallest
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing LexSmallestTerm
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing LexSmallestTerm
 -- internalChooseMinor: Choosing LexSmallestTerm
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing LexSmallest
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing LexSmallestTerm
 -- internalChooseMinor: Choosing LexSmallest
 -- internalChooseMinor: Choosing LexSmallestTerm
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing LexSmallest
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing LexSmallestTerm
 -- internalChooseMinor: Choosing LexSmallestTerm
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing LexSmallest
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing LexSmallest
 -- internalChooseMinor: Choosing LexSmallest
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing LexSmallestTerm
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing LexSmallest
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing LexSmallestTerm
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing LexSmallest
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing LexSmallest
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing LexSmallestTerm
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing LexSmallestTerm
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing LexSmallestTerm
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing LexSmallest
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing LexSmallest
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing LexSmallestTerm
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing LexSmallestTerm
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing LexSmallestTerm
 -- internalChooseMinor: Choosing LexSmallest
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing LexSmallest
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing LexSmallestTerm
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing LexSmallest
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing LexSmallest
 -- internalChooseMinor: Choosing LexSmallest
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing LexSmallestTerm
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing LexSmallestTerm
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing LexSmallestTerm
 -- internalChooseMinor: Choosing LexSmallest
 -- internalChooseMinor: Choosing LexSmallest
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing LexSmallest
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing LexSmallest
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing LexSmallest
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing LexSmallestTerm
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing LexSmallest
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing LexSmallest
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing LexSmallest
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing LexSmallest
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing LexSmallestTerm
 -- internalChooseMinor: Choosing LexSmallest
 -- internalChooseMinor: Choosing LexSmallestTerm
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing LexSmallest
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing LexSmallest
 -- internalChooseMinor: Choosing LexSmallestTerm
 -- internalChooseMinor: Choosing LexSmallest
 -- internalChooseMinor: Choosing LexSmallestTerm
 -- internalChooseMinor: Choosing LexSmallestTerm
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing LexSmallestTerm
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing LexSmallest
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing LexSmallest
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing LexSmallestTerm
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing LexSmallestTerm
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing LexSmallestTerm
 -- internalChooseMinor: Choosing LexSmallest
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing LexSmallestTerm
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing LexSmallest
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing LexSmallestTerm
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing LexSmallest
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing LexSmallestTerm
 -- internalChooseMinor: Choosing LexSmallest
 -- internalChooseMinor: Choosing LexSmallest
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing LexSmallestTerm
 -- internalChooseMinor: Choosing LexSmallest
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing LexSmallestTerm
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing LexSmallestTerm
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing LexSmallestTerm
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing LexSmallest
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing LexSmallestTerm
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing LexSmallestTerm
 -- internalChooseMinor: Choosing LexSmallest
 -- internalChooseMinor: Choosing LexSmallestTerm
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing LexSmallest
 -- internalChooseMinor: Choosing LexSmallest
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- used 9.1529s (cpu); 6.98653s (thread); 0s (gc)
regularInCodimension: ring dimension =3, there are 17325 possible 4 by 4 minors, we will compute up to 327.599 of them.
regularInCodimension: About to enter loop
regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 9, and computed = 9
regularInCodimension:  isCodimAtLeast failed, computing codim.
regularInCodimension:  partial singular locus dimension computed, = 1
regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 11, and computed = 10
regularInCodimension:  isCodimAtLeast failed, computing codim.
regularInCodimension:  partial singular locus dimension computed, = 1
regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 15, and computed = 14
regularInCodimension:  isCodimAtLeast failed, computing codim.
regularInCodimension:  partial singular locus dimension computed, = 1
regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 21, and computed = 20
regularInCodimension:  isCodimAtLeast failed, computing codim.
regularInCodimension:  partial singular locus dimension computed, = 1
regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 28, and computed = 27
regularInCodimension:  isCodimAtLeast failed, computing codim.
regularInCodimension:  partial singular locus dimension computed, = 1
regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 37, and computed = 31
regularInCodimension:  isCodimAtLeast failed, computing codim.
regularInCodimension:  partial singular locus dimension computed, = 1
regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 49, and computed = 38
regularInCodimension:  isCodimAtLeast failed, computing codim.
regularInCodimension:  partial singular locus dimension computed, = 1
regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 64, and computed = 49
regularInCodimension:  isCodimAtLeast failed, computing codim.
regularInCodimension:  partial singular locus dimension computed, = 1
regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 84, and computed = 60
regularInCodimension:  isCodimAtLeast failed, computing codim.
regularInCodimension:  partial singular locus dimension computed, = 1
regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 110, and computed = 76
regularInCodimension:  isCodimAtLeast failed, computing codim.
regularInCodimension:  partial singular locus dimension computed, = 1
regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 144, and computed = 94
regularInCodimension:  isCodimAtLeast failed, computing codim.
regularInCodimension:  partial singular locus dimension computed, = 1
regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 188, and computed = 122
regularInCodimension:  isCodimAtLeast failed, computing codim.
regularInCodimension:  partial singular locus dimension computed, = 1
regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 245, and computed = 157
regularInCodimension:  isCodimAtLeast failed, computing codim.
regularInCodimension:  partial singular locus dimension computed, = 1
regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 319, and computed = 202
regularInCodimension:  isCodimAtLeast failed, computing codim.
regularInCodimension:  partial singular locus dimension computed, = 1
regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 328, and computed = 203
regularInCodimension:  isCodimAtLeast failed, computing codim.
regularInCodimension:  partial singular locus dimension computed, = 1
regularInCodimension:  Loop completed, submatrices considered = 328, and computed = 203.  singular locus dimension appears to be = 1

The maximum number of minors considered can be controlled by the option MaxMinors. Alternatively, it can be controlled in a more precise way by passing a function to the option MaxMinors. This function should have two inputs; the first is minimum number of minors needed to determine whether the ring is regular in codimension n, and the second is the total number of minors available in the Jacobian. The function regularInCodimension does not recompute determinants, so MaxMinors or is only an upper bound on the number of minors computed.

i17 : time regularInCodimension(2, S, Verbose=>true, MaxMinors=>30)
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing LexSmallestTerm
 -- internalChooseMinor: Choosing LexSmallestTerm
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing LexSmallestTerm
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing LexSmallestTerm
 -- internalChooseMinor: Choosing LexSmallestTerm
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing RandomNonZero
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing LexSmallestTerm
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing Random
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing GRevLexSmallest
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing LexSmallestTerm
 -- internalChooseMinor: Choosing GRevLexSmallestTerm
 -- internalChooseMinor: Choosing RandomNonZero
 -- used 1.76098s (cpu); 1.32887s (thread); 0s (gc)
regularInCodimension: ring dimension =3, there are 17325 possible 4 by 4 minors, we will compute up to 30 of them.
regularInCodimension: About to enter loop
regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 9, and computed = 9
regularInCodimension:  isCodimAtLeast failed, computing codim.
regularInCodimension:  partial singular locus dimension computed, = 1
regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 11, and computed = 11
regularInCodimension:  isCodimAtLeast failed, computing codim.
regularInCodimension:  partial singular locus dimension computed, = 1
regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 15, and computed = 14
regularInCodimension:  isCodimAtLeast failed, computing codim.
regularInCodimension:  partial singular locus dimension computed, = 1
regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 21, and computed = 19
regularInCodimension:  isCodimAtLeast failed, computing codim.
regularInCodimension:  partial singular locus dimension computed, = 1
regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 28, and computed = 24
regularInCodimension:  isCodimAtLeast failed, computing codim.
regularInCodimension:  partial singular locus dimension computed, = 1
regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 30, and computed = 25
regularInCodimension:  isCodimAtLeast failed, computing codim.
regularInCodimension:  partial singular locus dimension computed, = 1
regularInCodimension:  Loop completed, submatrices considered = 30, and computed = 25.  singular locus dimension appears to be = 1

If you set the option VerifyNonRegular => true, then Macaulay2 will try to verify that the ring is not regular in codimension n. Turning this on means that when the set where the minors computed so far has codimension n, then it evaluates the matrix at the generic point of a minimal prime of that set. If that evaluated Jacobian matrix has too low of a rank, then one has verified that variety is not regular in codimemsion n. We consider the same example as above, but notice now the function returns false instead of true. This sometimes can be slower and sometimes can be faster.

i18 : time regularInCodimension(2, S, VerifyNonRegular=>true)
 -- used 1.64484s (cpu); 1.15145s (thread); 0s (gc)

o18 = false

This function has many options which allow you to fine tune the strategy used to find interesting minors. You can pass it a HashTable specifying the strategy via the option Strategy. See LexSmallest for how to construct this HashTable. The default strategy is StrategyDefault, which seems to work well on the examples we have explored. However, caution must be exercised, because, even in the examples above, certain strategies work well while others do not. In the Abelian surface example, LexSmallest works very well, while LexSmallestTerm does not even typically correctly identify the ring as nonsingular (this is because there are a small number of entries with nonzero constant terms, which are selected repeatedly). However, in our first example, the LexSmallestTerm is much faster, and Random does not perform well at all.

i19 : StrategyCurrent#Random = 0;

i20 : StrategyCurrent#LexSmallest = 100;

i21 : StrategyCurrent#LexSmallestTerm = 0;

i22 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
 -- used 0.557254s (cpu); 0.380646s (thread); 0s (gc)

o22 = true

i23 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
 -- used 0.592391s (cpu); 0.384652s (thread); 0s (gc)

o23 = true

i24 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
 -- used 0.463006s (cpu); 0.345639s (thread); 0s (gc)

o24 = true

i25 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
 -- used 0.445839s (cpu); 0.314235s (thread); 0s (gc)

o25 = true

i26 : StrategyCurrent#LexSmallest = 0;

i27 : StrategyCurrent#LexSmallestTerm = 100;

i28 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
 -- used 3.19852s (cpu); 2.29019s (thread); 0s (gc)

i29 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
 -- used 3.24184s (cpu); 2.25496s (thread); 0s (gc)

i30 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
 -- used 0.382478s (cpu); 0.256797s (thread); 0s (gc)

o30 = true

i31 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
 -- used 0.518387s (cpu); 0.390984s (thread); 0s (gc)

o31 = true

i32 : time regularInCodimension(1, S, Strategy=>StrategyRandom)
 -- used 1.98601s (cpu); 1.57106s (thread); 0s (gc)

o32 = true

i33 : time regularInCodimension(1, S, Strategy=>StrategyRandom)
 -- used 1.61101s (cpu); 1.28505s (thread); 0s (gc)

o33 = true

The minimum number of minors computed before checking the codimension can also be controlled by an option MinMinorsFunction. This is should be a function of a single variable, the number of minors computed. Finally, via the option CodimCheckFunction, you can pass the regularInCodimension a function which controls how frequently the codimension of the partial Jacobian ideal is computed. By default this is the floor of 1.3^k. Finally, passing the option Modulus => p will do the computation after changing the coefficient ring to ZZ/p.

The options PairLimit and SPairsFunction are passed directly to isCodimAtLeast. You can turn off internal calls to codim/dim, and only use isCodimAtLeast by setting UseOnlyFastCodim => true.

Ways to use regularInCodimension:

regularInCodimension(ZZ,Ring)

For the programmer

The object regularInCodimension is a method function with options.

The source of this document is in FastMinors.m2:1950:0.

regularInCodimension -- attempts to show that the ring is regular in codimension n

Description

See also

Ways to use regularInCodimension:

For the programmer