REVIEW
The existence of small defining polynomials for number fields, II
Number Theory · Submitted 6 Nov 2026 · 17 references
Pith's verdict: The unconditional odd-degree bound is real and the proof is transparent; the conditional cubic-optimality result is honestly labelled and should be read as conditional on an announced zero-free half-plane.
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Referee report
The paper
The paper claims
Every odd-degree number field admits an integral defining polynomial of height below $D^{1/2}$ — at most $D^{(n-1)/(2n-1)}$ — and, conditionally, cubic fields reach the optimal exponent $1/3$.
The reading
What carries the argument
The load-bearing object is the inverse different $d_K^{-1}$, the trace-dual lattice of the ring of integers $O_K$, together with the successive-minima inequality $1 \le \lambda_j(L)\,\lambda_{n+1-j}(L^*) \ll_n 1$ for a lattice and its Euclidean dual. Two norm constructions carry the argument: a short integral generator $\theta$ yields the monic polynomial $N_{K/\mathbb{Q}}(x-\theta)$, whose height is the product of its conjugates, while short elements $\alpha,\beta$ of the inverse different yield the integral polynomial $D_K \cdot N_{K/\mathbb{Q}}(\alpha x - \beta)$. The statistic $m(K)$, the maximum degree of a proper subfield, enters through a combinatorial lemma: any rational subspace of dimension larger than $m(K)$ contains a ratio $\beta/\alpha$ that generates $K$. For the totally real improvement, the trace-zero sublattice $\Lambda = \{\alpha \in d_K^{-1} : \operatorname{Tr}(\alpha)=0\}$ supplies the covariance identity $\operatorname{covol}(\Lambda) = 2^{-r_2}\sqrt{r_1 + 4r_2}\, D_K^{-1/2}$ and the product bound $A^m B^{n-m-1} \ll_n D_K^{-1/2}$ that balances its shortest vector length $A$ against the $k$-th minimum $B$. Under the zero-free half-plane, a prime-ideal counting lemma converts a suitably scaled degree-one prime $p$ into the substitution $f(x) \mapsto f(px+r)/p$, saving one factor of $p$ in the height.
Extended reading notes
Core claim
At the center stands Theorem 1: every number field $K$ of odd degree $n \ge 3$ and absolute discriminant $D_K$ possesses an integral polynomial $f(x) = a_0x^n + \cdots + a_n \in \mathbb{Z}[x]$ of degree $n$ with a root $\alpha$ such that $K = \mathbb{Q}(\alpha)$ and $H(f) = \max |a_i| \ll_n D_K^{(n-1)/(2n-1)}$. This sharpens the square-root bound $O_n(D_K^{1/2})$ that the authors' earlier work established for all degrees, and it is a genuine improvement because the exponent $1/2$ is known to be optimal in even degrees, the obstruction coming from towers of quadratic extensions and a relative discriminant inequality. The mechanism is lattice duality: a short integral generator produces a monic norm polynomial, whereas short elements of the inverse different produce the non-monic polynomial $D_K \cdot N_{K/\mathbb{Q}}(\alpha x - \beta)$, and the inequality relating successive minima of a lattice and its trace dual guarantees that one of the two constructions always succeeds; the real-embedding translation saves one factor, and the absence of an index-two subfield is what guarantees a generating ratio exists. Under a zero-free half-plane for the Dedekind zeta function, a degree-one prime at a suitable scale supplies the substitution $f(x) \mapsto f(px+r)/p$, which divides the height by $p$ and preserves the field, yielding the exponent $\rho(n,m(K))$ for fields of every signature, and in degree three the matching lower bound shows the optimal exponent is exactly $1/3$ along an infinite family of pure cubic fields.
Load-bearing premise
The only external load-bearing premise is the announced zero-free half-plane for Dedekind zeta functions (the hypothesis of Theorems 4 and 5, stated with parameter $N$ and currently quoted for $N = 32$); if it fails, the conditional theorems collapse, while the unconditional odd-degree bound of Theorem 1, which rests only on standard geometry-of-numbers inequalities, survives.
Editorial extensions
If this is right
- The 1998 square-root conjecture is not the end of the story in odd degrees: every field of degree $n \ge 3$ gets the unconditional exponent $(n-1)/(2n-1)$, which is $1/2 - 1/(4n-2)$ and therefore strictly below $1/2$.
- The improvement is a statement about subfield structure, not just parity: any field with a real embedding and no subfield of index two satisfies the same unconditional bound, transferring the result to non-odd signatures.
- Totally real fields do better, with exponent $\rho(n,m) = (n-1)/(2n-1 + 1/(2m-1))$ where $m$ is the largest proper subfield degree; for primitive fields this is $(n-1)/(2n)$, and every totally real cubic field already reaches the exponent $1/3$ unconditionally.
- Under the announced zero-free half-plane for Dedekind zeta functions, primitive fields of every signature attain exponent $(n-1)/(2n)$, and for cubic fields the exponent is optimal: $H(K) \ll_N D_K^{1/3}$ for all cubics, with $H(K) \ge \frac16 D_K^{1/3}$ for infinitely many.
- The optimal exponent by degree, Galois group, and real signature is now a well-posed open question: it is exactly $1/2$ in even degrees, exactly $1/3$ conditionally in degree three, strictly below $1/2$ in most other cases, and at least $1/3$ in every degree divisible by three because of the relative lower-bound family.
Reading between the lines
- Editorial inference: the proof's dual-lattice picture suggests the optimal height exponent is controlled by how the successive minima of $O_K$ and $d_K^{-1}$ interlace, so one might predict that fields with poorly balanced lattices are exactly the ones forced onto the larger exponents such as $1/2$ or $(n-1)/(2n-1)$.
- Editorial inference: if the zero-free hypothesis ever fails, the conditional theorems collapse but the unconditional Theorem 1 still leaves a gap of roughly $1/(4n)$ below $1/2$ in odd degrees, so the surviving content would be a uniform 'better than square-root' statement in odd degrees regardless of the behavior of zeta functions.
- Editorial inference: since the constants in Theorems 1–3 are effectively computable, the proof is in principle a construction; implementing it for cubic and quintic fields and comparing the minimal heights found with $D^{2/5}$ and $D^{4/9}$ would test whether the exponents are attained with usable constants.
- Editorial inference: the relative construction over a fixed base field suggests a template for other degrees — replacing the cubic root by a higher-degree obstruction might force analogous universal exponents in degrees not divisible by three, a direction the paper does not pursue.
Original abstract
We prove that every number field of odd degree $n\geq 3$ and absolute discriminant $D$ has an integral defining polynomial whose coefficients are $O_n(D^{(n-1)/(2n-1)})$ in size. This strengthens, in odd-degree cases, the recently established bound of $O_n(D^{1/2})$ for all degrees $n$, which was conjectured in 1998 by Ruppert and is now known to be optimal in even degrees. More generally, we prove that this strengthened bound holds for all degree-$n$ fields having a real embedding and no subfield of index two. The proof makes use of the lattice shapes of the ring of integers and its trace dual, the inverse different. Assuming the quasi-Riemann hypothesis for Dedekind zeta functions, as recently announced by Cureton and Thorner, we obtain an exponent smaller than $1/2$ for every field with no index-two subfield, of any signature. In the case of primitive fields, we obtain the exponent $(n-1)/(2n)$. In particular, we prove that the optimal exponent for cubic fields is $1/3$.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
We prove that every number field of odd degree $n\geq 3$ and absolute discriminant $D$ has an integral defining polynomial whose coefficients are $O_n(D^{(n-1)/(2n-1)})$ in size. This strengthens, in odd-degree cases, the recently established bound of $O_n(D^{1/2})$ for all degrees $n$, which was conjectured in 1998 by Ruppert and is now known to be optimal in even degrees. More generally, we prove that this strengthened bound holds for all degree-$n$ fields having a real embedding and no subfield of index two. The proof makes use of the lattice shapes of the ring of integers and its trace dual, the inverse different. Assuming the quasi-Riemann hypothesis for Dedekind zeta functions, as recently announced by Cureton and Thorner, we obtain an exponent smaller than $1/2$ for every field with no index-two subfield, of any signature. In the case of primitive fields, we obtain the exponent $(n-1)/(2n)$. In particular, we prove that the optimal exponent for cubic fields is $1/3$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We prove that every number field of odd degree $n\geq 3$ and absolute discriminant $D$ has an integral defining polynomial whose coefficients are $O_n(D^{(n-1)/(2n-1)})$ in size. | result 0.91 | 0.96 | 0.91 | asserted 1.00 |
| s1 | This strengthens, in odd-degree cases, the recently established bound of $O_n(D^{1/2})$ for all degrees $n$, which was conjectured in 1998 by Ruppert and is now known to be optimal in even degrees. | interpretation 0.40 | 0.16 | 0.12 | asserted 1.00 |
| s2 | More generally, we prove that this strengthened bound holds for all degree-$n$ fields having a real embedding and no subfield of index two. | result 0.93 | 0.94 | 0.12 | asserted 1.00 |
| s3 | The proof makes use of the lattice shapes of the ring of integers and its trace dual, the inverse different. | method 1.00 | 0.76 | 0.02 | asserted 1.00 |
| s4 | Assuming the quasi-Riemann hypothesis for Dedekind zeta functions, as recently announced by Cureton and Thorner, we obtain an exponent smaller than $1/2$ for every field with no index-two subfield, of any signature. | result 0.88 | 0.81 | 0.05 | asserted 0.70 |
| s5 | In the case of primitive fields, we obtain the exponent $(n-1)/(2n)$. | result 1.00 | 0.93 | 0.07 | asserted 1.00 |
| s6 | In particular, we prove that the optimal exponent for cubic fields is $1/3$. | result 0.99 | 0.94 | 0.13 | asserted 1.00 |
What was examined, and what was not
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References
Discussion
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