# compareFPT -- checks whether a given number is less than, greater than, or equal to the F-pure threshold

## Synopsis

• Usage:
compareFPT(t, f)
• Inputs:
• t,
• f, , an element of a -Gorenstein ring
• Optional inputs:
• FrobeniusRootStrategy => , default value Substitution, an option passed to computations in the TestIdeals package
• AssumeDomain =>
• MaxCartierIndex => an integer
• QGorensteinIndex => an integer
• Outputs:

## Description

This function returns -1 if t is less than the F-pure threshold of f. It returns 1 if t is greater than the F-pure threshold f. Finally, it returns 0 if it is equal to the F-pure threshold.

 `i1 : R = ZZ/7[x,y];` `i2 : f = y^2-x^3;` ```i3 : compareFPT(1/2, f) o3 = -1``` ```i4 : compareFPT(5/6, f) o4 = 0``` ```i5 : compareFPT(6/7, f) o5 = 1```

This function can also check the FPT in singular (but still strongly F-regular) ring, so long as the ring is also Q-Gorenstein of index dividing p-1. In the future we hope that this functionality will be extended to all Q-Gorenstein rings. In the following exam, x defines a Cartier divisor which is twice one of the rulings of the cone.

 `i6 : R = ZZ/5[x,y,z]/ideal(x*y-z^2);` `i7 : f = x;` ```i8 : compareFPT(1/3, f) o8 = -1``` ```i9 : compareFPT(1/2, f) o9 = 0``` ```i10 : compareFPT(13/25, f) o10 = 1```