Consider an element $f$ of a polynomial ring $R$ over a finite field of characteristic $p$, and an ideal $J$ of this ring. If $f$ is contained in the radical of $J$, then the command nu(e,f,J) outputs the maximal exponent $n$ such that $f^{ n}$ is not contained in the $p^e$-th Frobenius power of $J$. More generally, if $I$ is an ideal contained in the radical of $J$, then nu(e,I,J) outputs the maximal integer exponent $n$ such that $I^n$ is not contained in the $p^e$-th Frobenius power of $J$.
These numbers are denoted $\nu_f^J(p^e)$ and $\nu_I^J(p^e)$, respectively, in the literature, and were originally defined in the paper $F$-thresholds and Bernstein-Sato Polynomials, by Mustaţă, Takagi, and Watanabe.
i1 : R = ZZ/11[x,y]; |
i2 : I = ideal(x^2 + y^3, x*y); o2 : Ideal of R |
i3 : J = ideal(x^2, y^3); o3 : Ideal of R |
i4 : nu(1, I, J) o4 = 24 |
i5 : f = x*y*(x^2 + y^2); |
i6 : nu(3, f, J) o6 = 1330 |
If $f$ or $I$ is zero, then nu returns 0; if $f$ or $I$ is not contained in the radical of $J$, nu returns infinity.
i7 : nu(1, 0_R, J) o7 = 0 |
i8 : nu(1, 1_R, J) o8 = infinity o8 : InfiniteNumber |
When the third argument is omitted, the ideal $J$ is assumed to be the homogeneous maximal ideal of $R$.
i9 : R = ZZ/17[x,y,z]; |
i10 : f = x^3 + y^4 + z^5; |
i11 : nu(2, f) o11 = 220 |
i12 : nu(2, f, ideal(x, y, z)) o12 = 220 |
It is well known that if $q=p^e$ for some nonnegative integer $e$, then $\nu_I^J(qp) = \nu_I^J(q) p + L$, where the error term $L$ is nonnegative, and can be explicitly bounded from above in terms of $p$ and the number of generators of $I$ and $J$ (e.g., $L$ is at most $p-1$ when $I$ is principal). This implies that when searching for nu(e,I,J), it is always safe to start at $p$ times nu(e-1,I,J), and one need not search too far past this number, and suggests that the most efficient way to compute nu(e,I,J) is to compute, successively, nu(i,I,J), for i = 0,\ldots,e. This is indeed how the computation is done in most cases.
If $M$ is the homogeneous maximal ideal of $R$ and $f$ is an element of $R$, the numbers $\nu_f^M(p^e)$ determine and are determined by the $F$-pure threshold of $f$ at the origin. Indeed, $\nu_f^M(p^e)$ is $p^e$ times the truncation of the non-terminating base $p$ expansion of fpt($f$) at its $e$^{th} spot. This fact is used to speed up the computations for certain polynomials whose $F$-pure thresholds can be quickly computed via special algorithms, namely diagonal polynomials, binomials, forms in two variables, and polynomials whose factors are in simple normal crossing. This feature can be disabled by setting the option UseSpecialAlgorithms (default value true) to false.
i13 : R = ZZ/17[x,y,z]; |
i14 : f = x^3 + y^4 + z^5; -- a diagonal polynomial |
i15 : time nu(3, f) -- used 0.00631428 seconds o15 = 3756 |
i16 : time nu(3, f, UseSpecialAlgorithms => false) -- used 0.322419 seconds o16 = 3756 |
The valid values for the option ContainmentTest are FrobeniusPower, FrobeniusRoot, and StandardPower. The default value of this option depends on what is passed to nu. Indeed, by default, ContainmentTest is set to FrobeniusRoot if nu is passed a ring element $f$, and is set to StandardPower if nu is passed an ideal $I$. We describe the consequences of setting ContainmentTest to each of these values below.
First, if ContainmentTest is set to StandardPower, then the ideal containments checked when computing nu(e,I,J) are verified directly. That is, the standard power $I^n$ is first computed, and a check is then run to see if it is contained in the $p^e$-th Frobenius power of $J$.
Alternately, if ContainmentTest is set to FrobeniusRoot, then the ideal containments are verified using Frobenius Roots. That is, the $p^e$-th Frobenius root of $I^n$ is first computed, and a check is then run to see if it is contained in $J$. The output is unaffected, but this option often speeds up computations, specially when a polynomial or principal ideal is passed as the second argument.
i17 : R = ZZ/5[x,y,z]; |
i18 : f = x^3 + y^3 + z^3 + x*y*z; |
i19 : time nu(4, f) -- ContainmentTest is set to FrobeniusRoot, by default -- used 0.260487 seconds o19 = 499 |
i20 : time nu(4, f, ContainmentTest => StandardPower) -- used 3.57058 seconds o20 = 499 |
Finally, when ContainmentTest is set to FrobeniusPower, then instead of producing the invariant $\nu_I^J(p^e)$ as defined above, nu instead outputs the maximal integer $n$ such that the $n$^{th} (generalized) Frobenius power of $I$ is not contained in the $p^e$-th Frobenius power of $J$. Here, the $n$^{th} Frobenius power of $I$, when $n$ is a nonnegative integer, is as defined in the paper Frobenius Powers by Hernández, Teixeira, and Witt, which can be computed with the function frobeniusPower, from the TestIdeals package. In particular, nu(e,I,J) and nu(e,I,J,ContainmentTest=>FrobeniusPower) need not agree. However, they will agree when $I$ is a principal ideal.
i21 : R = ZZ/3[x,y]; |
i22 : M = ideal(x, y); o22 : Ideal of R |
i23 : nu(3, M^5) o23 = 10 |
i24 : nu(3, M^5, ContainmentTest => FrobeniusPower) o24 = 8 |
The function nu works by searching through the list of potential integers $n$ and checking containments of $I^n$ in the specified Frobenius power of $J$. The way this search is approached is specified by the option Search, which can be set to Binary (the default value) or Linear.
i25 : R = ZZ/5[x,y,z]; |
i26 : f = x^2*y^4 + y^2*z^7 + z^2*x^8; |
i27 : time nu(5, f) -- uses binary search (default) -- used 0.503509 seconds o27 = 1124 |
i28 : time nu(5, f, Search => Linear) -- used 0.674138 seconds o28 = 1124 |
i29 : M = ideal(x, y, z); o29 : Ideal of R |
i30 : time nu(2, M, M^2) -- uses binary search (default) -- used 3.40054 seconds o30 = 97 |
i31 : time nu(2, M, M^2, Search => Linear) -- but linear search gets luckier -- used 0.87433 seconds o31 = 97 |
The option AtOrigin (default value true) can be turned off to tell nu to effectively do the computation over all possible maximal ideals $J$ and take the minimum.
i32 : R = ZZ/7[x,y]; |
i33 : f = (x - 1)^3 - (y - 2)^2; |
i34 : nu(1, f) o34 = infinity o34 : InfiniteNumber |
i35 : nu(1, f, AtOrigin => false) o35 = 5 |
The option ReturnList (default value false) can be used to request that the output be not only $\nu_I^J(p^e)$, but a list containing $\nu_I^J(p^i)$, for $i=0,\ldots,e$.
i36 : R = ZZ/5[x,y,z]; |
i37 : f = x^2*y^4 + y^2*z^7 + z^2*x^8; |
i38 : nu(5, f, ReturnList => true) o38 = {0, 1, 8, 44, 224, 1124} o38 : List |
Alternatively, the option Verbose (default value false) can be used to request that the values $\nu_I^J(p^i)$ ($i=0,\ldots,e$) be printed as they are computed, to monitor the progress of the computation.
i39 : nu(5, f, Verbose => true) nuInternal: using comparison test FrobeniusRoot ν(1) = 0 ν(p^1) = 1 ν(p^2) = 8 ν(p^3) = 44 ν(p^4) = 224 ν(p^5) = 1124 o39 = 1124 |
The object nu is a method function with options.