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FrobeniusThresholds :: isFJumpingExponent

isFJumpingExponent -- whether a given number is an F-jumping exponent



Consider a $\mathbb{Q}$-Gorenstein ring $R$ of characteristic $p>0$, of index not divisible by $p$. Given an element $f$ of $R$ and a rational number $t$, isFJumpingExponent(t, f) returns true if $t$ is an $F$-jumping exponent of $f$, and otherwise it returns false.

i1 : R = ZZ/5[x,y];
i2 : f =  x^4 + y^3 + x^2*y^2;
i3 : isFJumpingExponent(7/12, f)

o3 = true
i4 : isFJumpingExponent(4/5, f)

o4 = true
i5 : isFJumpingExponent(5/6, f)

o5 = false
i6 : isFJumpingExponent(11/12, f)

o6 = true

The ring $R$ below is singular, and the jumping numbers of $f$ in the open unit interval are 1/4, 1/2 and 3/4.

i7 : R = ZZ/11[x,y,z]/(x*y - z^2);
i8 : f = x^2;
i9 : isFJumpingExponent(1/4, f)

o9 = true
i10 : isFJumpingExponent(3/8, f)

o10 = false
i11 : isFJumpingExponent(1/2, f)

o11 = true
i12 : isFJumpingExponent(2/3, f)

o12 = false
i13 : isFJumpingExponent(3/4, f)

o13 = true

Setting the option AtOrigin to true (its default value is false) tells the function to consider only $F$-jumping exponents at the origin. The following example considers a polynomial that looks locally analytically like two lines at the origin, and four lines at (2,0).

i14 : R = ZZ/13[x,y];
i15 : f = y*((y + 1) - (x - 1)^2)*(x - 2)*(x + y - 2);
i16 : isFJumpingExponent(3/4, f)

o16 = true
i17 : isFJumpingExponent(3/4, f, AtOrigin => true)

o17 = false

If the ambient ring $R$ is not a domain, the option AssumeDomain should be set to false. We assume that the ring is a domain by default, in order to speed up the computation.

If the Gorenstein index of $R$ is known, the user should set the option QGorensteinIndex to this value. Otherwise, the function attempts to find the Gorenstein index of $R$, assuming it is between 1 and the value passed to the option MaxCartierIndex (default value 10).

The option FrobeniusRootStrategy is passed to internal calls of functions from the TestIdeals package. The two valid values of FrobeniusRootStrategy are Substitution and MonomialBasis.

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

See also

Ways to use isFJumpingExponent :

For the programmer

The object isFJumpingExponent is a method function with options.