isSimpleNormalCrossing -- whether a polynomial is a product of factors that are in simple normal crossing

Synopsis

Usage:

isSimpleNormalCrossing(f)

isSimpleNormalCrossing(P)
Inputs:
- f, a ring element, a polynomial
- P, a product expression, a product of polynomials
Optional inputs:
- AtOrigin => a Boolean value, default value true, tells the function whether to consider only the behavior at the origin
- Verbose => a Boolean value, default value false, whether the output is to be verbose
Outputs:
- a Boolean value, true if the factors of f are in simple normal crossing, false otherwise

Description

Let $f$ be a polynomial or an object of class Product representing a factorization of a polynomial $f$. The function isSimpleNormalCrossing returns true if the distinct factors of $f$ are in simple normal crossing, or in other words, if at each point those factors, locally analytically, form part of a regular system of parameters.

i1 : R = ZZ/7[x,y,z];

i2 : isSimpleNormalCrossing(x^3*y^2)

o2 = true

i3 : isSimpleNormalCrossing(x^2 - y^2)

o3 = true

i4 : isSimpleNormalCrossing(x*y*(x - y))

o4 = false

i5 : isSimpleNormalCrossing(x^2 - y*z)

o5 = false

The option AtOrigin (default value true) is used to specify whether to work at the origin (when true) or globally (when false).

i6 : R = QQ[x,y,z];

i7 : f = (y - (x - 1)^2)*y^2; --SNC at the origin, but not globally

i8 : isSimpleNormalCrossing(f)

o8 = true

i9 : isSimpleNormalCrossing(f, AtOrigin => false)

o9 = false

i10 : g = (y - 1)^2 + (x - 7)^2; --doesn't even pass through the origin

i11 : isSimpleNormalCrossing(g)

o11 = true

i12 : isSimpleNormalCrossing(g, AtOrigin => false)

o12 = false

i13 : h = x*y^2*(x + 1)^3*(y - 1)^4; --SNC everywhere

i14 : isSimpleNormalCrossing(h)

o14 = true

i15 : isSimpleNormalCrossing(h, AtOrigin => false)

o15 = true

Setting the option Verbose (default value false) to true produces verbose output.

Ways to use isSimpleNormalCrossing :

isSimpleNormalCrossing(Product)
isSimpleNormalCrossing(RingElement)

For the programmer

The object isSimpleNormalCrossing is a method function with options.