Multivariable overconvergent Weierstrass division #
Let K be a complete ultrametric normed field, except where noted below. A series in K⟨X, Y⟩†,
with Y = X none, which is distinguished of degree s in Y at unit radius and has a unit
coefficient in degree s, divides every overconvergent series. The quotient is overconvergent in
all variables and the remainder is a polynomial in Y of degree less than s, with coefficients
in K⟨X⟩†. The decomposition is unique, and the quotient q and the coefficients of the
remainder r are bounded by the dividend f: ‖q‖ ‖g‖ ≤ ‖f‖ and ‖r.coeff i‖ ≤ ‖f‖.
Uniqueness, and the statements deduced from it alone (the comparison of divisibility, descent of
irreducibility, and the implication from primality in K⟨X, Y⟩† to primality in K⟨X⟩†[Y]), need
only that K is an ultrametric normed commutative ring with multiplicative norm. All other results,
including existence, the norm bounds, the quotient identification, the converse direction of
primality and the integral-quotient results, assume that K is a complete ultrametric normed
field.
Existence uses unequal radii: first select 1 < ρ ≤ r preserving distinguishedness, then
apply fixed-radius division over the Banach ring K{X/ρ} at radius r in Y. The quotient
is restricted at these polyradii, hence overconvergent. Uniqueness is checked in the
unit-radius Tate algebra. These are the division statements used in the ideal theory of
Monsky–Washnitzer algebras.
For a distinguished monic polynomial ω ∈ K⟨X⟩†[Y], evaluated at Y by
washnitzerAlgebra.polynomialToOption, uniqueness of division shows that divisibility by ω is
the same in K⟨X⟩†[Y] and in K⟨X, Y⟩†, and existence shows that every series is congruent to
a polynomial modulo ω. Together these identify K⟨X⟩†[Y] ⧸ (ω) with K⟨X, Y⟩† ⧸ (ω)
(washnitzerAlgebra.polynomialQuotientEquiv), so that adjoining a root of a monic polynomial
to K⟨X⟩† is presented by one more overconvergent variable. Hence also irreducibility of ω
descends from K⟨X, Y⟩† to K⟨X⟩†[Y], and ω is prime in one ring iff it is prime in the other.
Dividing one variable at a time by its minimal polynomial over K, a polynomial with constant
coefficients and hence distinguished at unit radius, shows that every quotient of K⟨X⟩† in
finitely many variables which is integral over K is spanned by the images of polynomials
(TauCeti.Dagger.washnitzerAlgebra.surjective_mkₐ_comp_ofMvPolynomial). Consequently two
algebra homomorphisms from K⟨X⟩† into an algebra integral over K that agree on the variables
are equal (TauCeti.Dagger.washnitzerAlgebra.algHom_ext_of_isIntegral).
References #
- E. Grosse-Klönne, Rigid analytic spaces with overconvergent structure sheaf, J. reine angew. Math. 519 (2000), §1.3(i).
- Bosch, Güntzer, Remmert, Non-Archimedean Analysis, §5.2.1, Theorem 2.
The fixed-radius division is TauCeti.PowerSeries.IsMulDistinguished.exists_mul_add_eq;
the radius selection is TauCeti.Dagger.exists_isDistinguished_isUnit_of_isDistinguished_one.
The polynomial comparisons follow their Tate-algebra counterparts
TauCeti.PowerSeries.IsDistinguished.polynomialToRestricted_dvd_iff and
TauCeti.PowerSeries.IsDistinguished.polynomialQuotientEquiv.
Uniqueness in overconvergent Weierstrass division. If q * g + r = q' * g + r'
with polynomial remainders r, r' of degree less than s in the separated variable, then
q = q' and r = r'. Neither completeness nor a unit distinguished coefficient is needed.
Divisibility of polynomials by a distinguished monic polynomial ω is unchanged on passing
to K⟨X, Y⟩†. This holds without assuming K complete.
A distinguished monic polynomial which is irreducible in K⟨X, Y⟩† is irreducible in
K⟨X⟩†[Y].
If a distinguished monic polynomial is prime in K⟨X, Y⟩†, it is prime in K⟨X⟩†[Y]. This
holds without assuming K complete.
Overconvergent Weierstrass division. If g is distinguished of degree s at unit
radius with unit distinguished coefficient, then every overconvergent dividend f has an
overconvergent quotient q and a polynomial remainder r over the smaller overconvergent
algebra, of degree less than s in the separated variable, with f = q * g + r.
Norm bounds in overconvergent Weierstrass division. Under the hypotheses of
exists_mul_add_eq_of_isDistinguished, the division f = q * g + r satisfies ‖q‖ ‖g‖ ≤ ‖f‖,
and every coefficient of the remainder r has Gauss norm at most ‖f‖.
Unique overconvergent Weierstrass division, with the quotient and polynomial remainder bundled as a unique pair. No characteristic or nontrivial-norm assumption is required.
Every overconvergent series is congruent modulo a distinguished monic polynomial ω to a
polynomial in Y of degree less than that of ω, with coefficients in K⟨X⟩†.
Quotients by a distinguished monic polynomial. Evaluation at Y identifies
K⟨X⟩†[Y] ⧸ (ω) with K⟨X, Y⟩† ⧸ (ω) for a distinguished monic polynomial ω. Its inverse
sends the class of a series to the class of its Weierstrass remainder.
Instances For
The quotient comparison sends the class of a polynomial to the class of its evaluation
at Y.
A distinguished monic polynomial is prime in K⟨X⟩†[Y] iff it is prime in K⟨X, Y⟩†.
Polynomials surject onto integral quotients. If a quotient of the Monsky–Washnitzer
algebra in finitely many variables is integral over K, for instance finite-dimensional, then
every class has a polynomial representative.
Homomorphisms into integral algebras are determined by the variables. Two K-algebra
homomorphisms from the Monsky–Washnitzer algebra in finitely many variables into a K-algebra
integral over K, for instance a finite-dimensional one, agree as soon as they agree on every
variable. No continuity is assumed.