Thanks to Xavier Caruso and David Lubicz, Magma now contains an implementation of linear algebra over (Z)p[[u]] and related rings [CL12]. This is a first step in various calculations with p-adic Hodge theory, to be able to compute with modules over suitable approximations in these rings.
The package deliberately builds a wrapper level through user-defined types in Magma. While one could alternatively just do everything in terms of power (and Laurent) series rings over the p-adics directly, this extra layer is hoped to make it easier to use the functionality in a coherent way. Some of the common functions applicable to power series, matrices, and modules have been copied over, but not all of them. The underlying attributes can be accessed directly in the cases where there is no equivalent on the types described here. Most notably, vectors and matrices that are user-defined types do not seem yet to be accesible via v[i] or M[i, j].
The package has seen limited testing, with not every possibility of precision bounds (in both the p-adic and power series aspects) being considered in all cases, nor has much work been done for base rings that are extensions of (Z)p.
Let R be a discrete valuation ring with residue field of characteristic p, and let ν be real. Recall that f(u)=∑i aiui where ai∈(Frac)(R) converges for |u|<1/pν if and only if infi vR(i) + iν is bounded.
When ν is rational, this infinimum is attained. It is called the Gauss valuation, denoted by vν(f), and the Weierstrass degree dν(f) is the minimal i that realises this (by convention, the Weierstrass degree of the zero element is -∞). An element is distinguished if its Gauss valuation is 0. We have that vν(fg)=vν(f) + vν(g) and vν(f + g)≥min(vν(f), vν(g)), while dν(fg)=dν(f)dν(g).
As it is not particularly clear how to represent elements of a convergent series ring in finite form, so we instead work with the "slope" ring of elements of nonnegative Gauss valuation for which vν(f)≥0, namely Sν(R)=biggl{∑i=0^∞ai ui| vR(ai) + i.ν≥0, ai∈(Frac)(R)biggr}. For instance, when ν=0, this ring is just the power series ring over R.
There are two related rings of interest. The first extends the coefficients to allow them to have negative valuation in (Frac)(R), namely Spν(R)=Sν[1/π] where π uniformises R. The second is a type of Laurent extension, namely writing ν=a/b in lowest terms, we have Suν(R)=Sν[πa/ub].
For instance, the polynomial u2/p is in S1/2((Q)p) and Sν((Q)p) for all ν, and inverting it gives us p/u2∈S1/2u((Q)p). Note that as a power series u2/p clearly has infinite radius of convergence, even though it is in not in the slope ring S0((Q)p).
Another example, p + u has Gauss valuation min(1, ν), and thus the Weierstrass degree is 0 for ν≥1 but degree 1 for ν<1.
Both of these "completion" rings are Euclidean, which allows us to construct a module theory over them. While internally Magma has generic code for handling modules over Euclidean rings, with these user-defined types the component parts are all taken from boilerplate package-level implementations.
Here are the basic creations and operations for Sν rings and elements. All of these have a direct analogue for Spν and Suν rings.
Precision: RngIntElt Default: ∞
Given a p-adic field (or ring) F and a slope ν (which can be rational or integral), create the slope ring (or the Spν or Suν ring). The precision refers to that of the power series ring.
nu: FldRatElt Default: 0
Precision: RngIntElt Default: ∞
The first three are the same as above, but with ν as a parameter. The others create the slope ring over (Q)p, with e being the precision of the p-adic ring in this case.
Given a power-series ring over a p-adic ring or field and a rational number (or integer), create the slope ring. For the Sνu ring, one gives a Laurent series ring.
Given an Sνp-ring, create the associated Sν-ring, and vice-versa. Similarly with an Sνu-ring.
Two slope rings are equal if their underlying p-adic power series rings are the same (including precisions), and the slopes ν are the same.
The slope of the Sν ring.
The precision of the underlying power-series ring.
The underlying p-adic ring of the Sν ring.
The full range of addition, subtraction, multiplication, division (where applicable) and powering are available, similarly with equality.
Return the Sν ring to which the element belongs.
This is the proper way to test equality with elements of Sν. It checks whether every (known) coefficient is zero to within the p-adic precision.
The Gauss valuation and Weierstrass degree respectively return these values.
The IsDistinguished intrinsic returns whether the Gauss valuation is zero (this intrinsic is only for Sν elements).
Gives the first non(weakly)zero term in the Sνp-element, as a power series (or Laurent series). Useful in particular when examining vectors or matrices of such.
Gives the term corresponding to the Weierstrass degree of the Sνp-element, as a power series (or Laurent series). Useful in particular when examining vectors or matrices of such.
Gives the big-Oh value of the argument.
Here are some basic operations with a ring of slope 1.
> S<u> := SnuRing (pAdicField (5, 15), 1); // slope 1 over 5-adics > assert Slope(S) eq 1; > f := 5*u + u^2 + 2*u^3 + u^5 + O(u^10); > g := u^4/25 + u^6 + u^8 + O(u^10); > GaussValuation (f); // both have Gauss valuation 2 2 > GaussValuation (g); // both have Gauss valuation 2 2 > q := g/f; q; // g has degree 4, so this division works (5^-3 + O(5^12))*u^3 + (-5^-4 + O(5^11))*u^4 + (616*5^-5 + O(5^10))*u^5 + (-606*5^-6 + O(5^9))*u^6 + (9946*5^-7 + O(5^8))*u^7 + (-3761*5^-8 + O(5^7))*u^8 + O(u^9) > LeadingTerm (q); // returned as a power series ring element (5^-3 + O(5^12))*$.1^3
All the same functionality copies over to the Sνp rings, the main difference is that one can invert p directly there. There is automatic coercion from Sν to Sνp rings in some cases, even when the slopes are not equal. The criterion is whether the result in the Sνp ring is valid.
> Sp<uu> := SpRing (pAdicField (5, 15), 1/2); // slope 1/2 over 5-adics > assert Slope(Sp) eq 1/2; > ff := 1/5 + uu + 5*uu^2 + O(uu^10); > GaussValuation (ff); -1 > gg := uu + 5^2*uu^2 + uu^3 - uu^6 + uu^9 + O(uu^10); > h := S ! (ff * S!gg); // coerced back to the Snu-ring > assert IsDistinguished (h); > LeadingTerm (ff) * LeadingTerm (gg) eq LeadingTerm (h); true
Similarly with Sνu rings, where Laurent series are now also allowed.
> Su<z> := SuRing (pAdicField (7, 25), 2/3); // slope 2/3 over 7-adics > F := 7/z + 3*7 + z^2/7 + 7^2*z^3 + O(z^10); > GaussValuation (F); 1/3 > WeierstrassTerm (F); (7 + O(7^26))*$.1^-1 > SR<t> := SnuRing (Su); > t*z/7; // automatic coercion into Su (7^-1 + O(7^24))*z^2 + O(z^20)
Given A, B∈Sν with vν(A)≥vν(B) (so in particular B is nonzero) the quotient-remainder algorithm returns Q, R such that A=BQ + R with R a polynomial of degree less that dν(B). In general the precision loss in this operation becomes more acute with larger ν. This precision loss percolates throughout all computations. Note that the result in general depends on ν.
The Weierstrass preparation theorem takes a distinguished element f∈Sν and writes f=UP where U is invertible in Sν(R) and P is a polynomial of degree dν(f). The units of Sν are precisely those elements with Gauss valuation and Weierstrass degree both 0.
The ring Sν is not itself Euclidean, as we can only properly quotient elements as A/B when vν(B)≥vν(A), and this property need not be preserved in the Euclidean steps. For instance with ν=1, starting with A=pu2 + p4 and B=p2u we get Q=u/p and R=p4, but then vν(R)=4>3=vν(B), so that we cannot divide B by R.
However, the ability to scale by arbitrary powers of the uniformiser implies that the ring Spν is Euclidean with respect to the Weierstrass degree, and similarly the ring Suν is a discrete valuation ring (hence Euclidean) for the valuation vν.
The extended gcd in Sν or Sνp takes A, B with v(A)≥v(B) and returns (G, H, w, x, y, z) has Aw + Bx=G where v(B)=v(G) and Ay + Bz=H with v(H)>v(A) with wz - xy=1 (consecutive Euclidean steps). The gcd is G in the case where H is the zero polynomial (this will always be true over Sνp).
Since Sνu is a discrete valuation ring, the quotient-remainder A=qB + r always has r=0 when v(B)<v(A), and else q=0. The canonical element of valuation v in Sνu is uy pz where 0≤y<(denom)(ν) is as small as possible and yν + z=v. The gcd of two elements A and B can always be taken to be a canonical element of valuation equal to the smaller of the valuations of A and B.
Given a distinguished element f of Sν (one of Gauss valuation 0), write f=UP where U is invertible in Sν and P is a polynomial (whose degree equals the Weierstrass degree of f).
Given Sν elements A and B with v(A)≥v(B), determine Q, R with A=BQ + R where B is a polynomial whose degree is less than the Weierstrass degree of B.
Given A and (nonzero) B which are Spν elements, determine Q, R with A=BQ + R where B is a polynomial whose degree is less than the Weierstrass degree of B.
Given A and (nonzero) B which are Suν elements, determine Q, R with A=BQ + R. Here when v(A)≥v(B) we have Q=A/B and R=0, and when v(A)<v(B) we have Q=0 and R=A.
Given Sν elements A and B with v(A)≥v(B), try to compute the gcd. This returns a 6-tuple (G, H, w, x, y, z) with Aw + Bx=G, Cw + Dy=H, and wz - xy=1. When H=0 then G is the gcd. Alternatively, v(H) will be some element with v(H)>v(A), which precludes the Euclidean algorithm from progressing.
Given Spν elements A and B compute the extended gcd. This returns a 6-tuple (G, H, w, x, y, z) with Aw + Bx=G, Cw + Dy=H=0, and wz - xy=1, where G is the gcd.
Given Suν elements A and B compute the extended gcd. Since Suν is a DVR, the gcd can be taken to be the CanonicalElement of the smaller valuation of A and B.
The canonical element of valuation v in Sνu is uy pz where 0≤y<(denom)(ν) is as small as possible and yν + z=v.
Here are some basic examples of Euclidean operations.
> S<u> := SnuRing (pAdicField (5, 15), 1); // slope 1 over 5-adics > f := 5*u + u^2 + 2*u^3 + u^5 + O(u^10); > g := u^4/25 + u^6 + u^8 + O(u^10); > q := g/f; > assert IsDistinguished (q); > U, P := WeierstrassPreparation (q); // loses a lot of precision > U; 1 + O(5^15) + (-5^-1 + O(5^14))*u + (616*5^-2 + O(5^13))*u^2 + (-606*5^-3 + O(5^12))*u^3 + (9946*5^-4 + O(5^11))*u^4 + (-3761*5^-5 + O(5^10))*u^5 + O(u^6) > P; (5^-3 + O(5^12))*u^3 + O(u^6) > q, r := Quotrem (g, f); assert r eq 0; assert q eq g/f; > q; // same as above, though is O(u^10) not O(u^9) (5^-3 + O(5^12))*u^3 + (-5^-4 + O(5^11))*u^4 + (616*5^-5 + O(5^10))*u^5 + (-606*5^-6 + O(5^9))*u^6 + (9946*5^-7 + O(5^8))*u^7 + (-3761*5^-8 + O(5^7))*u^8 + (-172699*5^-9 + O(5^6))*u^9 + O(u^10) > q, r := Quotrem (f, g); // this is valid, as the valuations are equal > q, r; assert f eq g*q + r; (5^2 + O(5^17))*u + (-5^4 + O(5^17))*u^3 + (24*5^4 + O(5^17))*u^5 + (-23*5^6 + O(5^17))*u^7 + (551*5^6 + O(5^17))*u^9 + O(u^10) (5 + O(5^16))*u + (1 + O(5^15))*u^2 + (2 + O(5^15))*u^3 + O(u^10) > G, H, A, B := ExtendedGcd (f, g); assert H eq 0; > G; (-5 + O(5^16))*u + (-1 + O(5^15))*u^2 + (-2 + O(5^15))*u^3 + O(u^10) > assert A*f + B*g eq G;
> f1 := 5*u^2 + 5^4; // example from the text > g1 := 5^2*u; // where ExtendedGcd fails > G, H := ExtendedGcd (f1, g1); assert H ne 0; > H; -5^4 + O(5^24) + O(u^20) > GaussValuation (H); GaussValuation (g1); 4 3
The above will also work in Sνp. As noted above, the corresponding commands for Sνu rings are particularly simplified.
> Su<z> := SuRing (S); > F := Su!f; assert GaussValuation(F) eq 2; > G := Su!g; assert GaussValuation(G) eq 2; > Q, R := Quotrem (F, G); assert R eq 0; > g, _, A, B, C, D := ExtendedGcd (F, G); > g; assert g eq CanonicalElement (Su, 2); 5^2 + O(5^17) + O(z^20) > assert A*F + B*G eq g; > assert C*F + D*G eq 0; > assert A*D - B*C eq 1;