Countable semiring partitions of real function fields
September 8, 2026
Abstract
For every real transcendental number , we construct a partition of into countably infinitely many nonempty sets closed under addition and multiplication. The construction answers the countable-decomposition question of Kiss, Somlai, and Terpai. A rational function is assigned the nearest zero or pole to the left of , together with the distinction between a zero and a pole. Positivity on the intervening interval makes these labels stable under addition and multiplication within a class. A version using both endpoints separates all the positive affine functions proposed in that question. The same endpoint argument on smooth real curves proves that the positive elements of every finitely generated subfield of of positive transcendence degree admit such a partition.
1. Introduction
Call a nonempty subset of an ordered field invariant if it is closed under addition and multiplication. We use “semiring partition” for a partition into invariant sets; the pieces are not required to contain zero or a multiplicative identity.
Kiss, Somlai, and Terpai study finite invariant partitions of positive elements of real fields [1]. For a real transcendental , their two-class classification is controlled by the sign of the derivation on induced by formal differentiation. In Section 7, pp. 530–531, they ask whether , or even , admits a partition into countably infinitely many invariant sets. They also ask whether such a partition can separate the positive affine functions associated with distinct rational roots. We give an explicit construction with these properties.
Theorem 1.
Let be a countable subfield of , and let be transcendental over . Both and have partitions into countably infinitely many invariant sets. The partitions can be chosen so that the elements
lie in distinct pieces for distinct .
The argument requires only finitely many real zeros and poles and elementary one-sided limits. Its extension to curves gives the following further result.
Theorem 2.
Let be a finitely generated field extension of with positive transcendence degree. Then has a partition into countably infinitely many invariant sets.
In particular, the conclusion holds for every finite real algebraic extension of , with no restriction on the genus of its function field. Theorem 1 has a direct proof independent of the curve argument. Neither theorem classifies finite partitions, and the finite-class conjecture in [1, Section 7] is not addressed here.
2. The nearest endpoint construction
Evaluation at identifies with injectively. Every nonzero rational function in is finite and nonzero at : a nonzero numerator or denominator over cannot vanish there. Write
Zeros and poles always refer to the rational function after cancellation, not to a chosen unreduced expression.
For , let be its largest real zero or pole strictly below , if one exists. Put otherwise. There are finitely many such points. The function is finite, continuous, and positive throughout , since it has no zero or pole there and is positive at .
If is finite, label it when it is a zero and when it is a pole. Equivalently, the right-hand limit of there is respectively or . Define
Lemma 3.
If for , then
Proof.
Suppose first that the common endpoint is a finite real number . Both and are finite and strictly positive on . Their sum and product therefore have the same property. Consequently neither has a zero or a pole in that interval.
If the common type is , then both functions tend to zero as . Their sum and product also tend to zero. A nonzero rational function with this limit has a zero at after cancellation. If the common type is , both functions tend to as . The same holds for their sum and product, which therefore have poles at . Thus remains the nearest zero or pole to the left, with its type unchanged.
If the label is , both functions are finite and strictly positive on . Their sum and product are finite and strictly positive there too, so both labels remain .
Corollary 4.
The nonempty fibres of form a countably infinite invariant partition of . Its restriction to positive polynomials is also countably infinite.
Proof.
The fibres of a function are pairwise disjoint and cover its domain. Lemma 3 gives invariance. Since is countable, is countable, so there are at most countably many nonempty fibres. For every with , the polynomial has label . There are infinitely many such , already among the rationals. These are distinct nonempty fibres, both for rational functions and for polynomials.
Remark 5.
The distinction between zeros and poles is necessary. For , the functions and have the same nearest endpoint but their product is , which has no endpoint. Exact vanishing orders should not be part of the label either: multiplication adds the orders. Their signs suffice, because positive one-sided values prevent cancellation of the leading terms in a sum of two poles.
3. Both endpoints and affine generators
For , let be the connected component containing of the set of real points where is finite and strictly positive. It is an open interval , allowing infinite endpoints. Every finite endpoint is a zero or a pole. Label by this interval and the zero or pole type at each finite endpoint. Write for this label.
Proposition 6.
The nonempty fibres of are invariant. They give the partitions in Theorem 1.
Proof.
If , both functions are finite and strictly positive throughout the same interval . Their sum and product are finite and strictly positive on . At each finite endpoint the limits taken from within are either both zero or both . The argument of Lemma 3, applied separately at each endpoint, shows that the same endpoint and type persist for both operations. The positive component containing cannot pass such an endpoint. Thus its interval is exactly and its types are unchanged. An infinite endpoint requires no additional condition.
There are at most countably many nonempty fibres. For , the function has interval , with a zero at its finite endpoint. For , the function has interval , again with a zero at its finite endpoint. These labels are pairwise distinct as varies. They prove infinitude, the claimed separation, and the polynomial assertion. Evaluation at transfers the partitions to the numerical fields and rings in the theorem.
The construction also contains the invariant sets suggested by affine generators. For any and any nonzero polynomial
the function is positive exactly throughout the indicated half-line adjoining the root , up to possible further positive components on its other side. Its component containing is that half-line, and its limit at is zero. Hence all these functions lie in the fibre of . Allowing a positive constant term would remove the zero at and is not part of this assertion.
The two endpoint labels refine the left endpoint labels, but do not refine the derivative-sign partition. For example, and are positive and finite on all of , so have the same label . Their derivatives at a transcendental have opposite signs. All positive constants belong to the same fibre as these functions.
4. The construction on real curves
We first state the geometric version. A point on a curve defined over is generic over here if its morphism has image the generic point of . Equivalently, evaluation at embeds the function field into ; every nonzero -rational function is finite and nonzero there.
Proposition 7.
Let be countable. Let be a smooth projective integral curve over , and let be generic over . Ordered by evaluation at , the positive elements of admit a countably infinite invariant partition.
Proof.
The real points of a smooth projective curve form a compact smooth one-dimensional manifold, by the implicit function theorem in affine charts and compactness of real projective space. Let be the connected component containing ; it is a circle. This remains true if the base change of to is disconnected. We use only its component through .
For a nonzero , the restriction to is real meromorphic and has only finitely many zeros and poles. Locally it has the form , where is a real analytic parameter, , and is analytic and nonzero. The divisor of has finite support; this remains a finite set after base change. No such point is by genericity.
If , let be the connected component containing of the set where is finite and positive on . Either , or is an open arc. Its boundary consists of at most two points, each a zero or a pole. The two ends of an arc may correspond to the same point of the circle; the arc itself, not just an unordered pair of points, is retained in the label. Label by and the zero or pole type at each boundary point, or by if there is no boundary.
If two functions have the same label, their sum and product are finite and strictly positive on the common component. At every end they have the same limiting behaviour, either zero or , from within that component. These limits persist for both operations. The component cannot enlarge past a boundary point and cannot acquire a new boundary inside. Thus the label is unchanged. This proves invariance, including the case .
The field is countable, so the number of nonempty fibres is at most countable. To prove infinitude, choose a function transcendental over . It gives a nonconstant finite morphism . In characteristic zero this morphism is generically unramified. Its poles and ramification points form a finite set defined over , and genericity excludes from that set. Thus is a real local coordinate near .
Choose a small arc about on which is strictly monotone. For every rational sufficiently close to , there is a unique point on this arc with . The function is positive on the subarc between and , and has a zero at . Therefore is a boundary point of . These points are distinct for distinct . Each labelled arc has at most two boundary points, so infinitely many distinct labels occur. This proves countable infinitude.
Choose a transcendence basis for , with , and put , taking when . Then is countable and is a finitely generated extension of transcendence degree one. It has a nonsingular projective integral curve model over , which is smooth because has characteristic zero; see [2, Theorem 53.2.6 and Lemma 53.2.8]. Identify its function field with .
The given embedding defines a morphism , and hence a real point generic over . Evaluation at this point is the original embedding of , so the order in Proposition 7 is the prescribed order. That proposition gives the required partition of .
5. Scope and further questions
The construction proves countable infinitude directly; it does not arise by repeatedly splitting one of the finite derivative-sign partitions. Indeed, its whole-line class can contain elements of opposite derivative signs. This explains why it places no constraint on the finite-class classification problem.
The endpoint method is stable under passage to smooth projective models and hence permits finite algebraic extensions of a rational function field. It does not, by itself, provide compatible partitions along an infinite algebraic tower. Determining whether every countable transcendental subfield of admits a countably infinite invariant partition is a further question. Nor do we claim that the endpoint partitions classify all countable invariant partitions. The result proved here is the existence statement and its extension to finitely generated real fields.
Preparation note.
This manuscript was prepared with OpenAI Codex assistance in problem selection, proof development, source comparison, and writing. The accompanying repository records internal checks. No independent human review or journal acceptance is claimed.
References
[1] G. Kiss, G. Somlai, and T. Terpai, Decompositions of the positive real numbers into disjoint sets closed under addition and multiplication, Journal of Algebra 664 (2025), 511–533. doi:10.1016/j.jalgebra.2024.10.044.
[2] The Stacks Project Authors, The Stacks Project, Section 53.2, Curves and function fields, Theorem 53.2.6 (Tag 0BY1) and Lemma 53.2.8. https://stacks.math.columbia.edu/tag/0BXX. Accessed September 8, 2026.