Linear independence of powers

Andrew Snowden; Steven V Sam

arxiv: 1907.02659 · v1 · pith:PTLZYWHEnew · submitted 2019-07-05 · 🧮 math.AC

Linear independence of powers

Steven V Sam , Andrew Snowden This is my paper

Pith reviewed 2026-05-25 02:14 UTC · model grok-4.3

classification 🧮 math.AC

keywords linear independencepowers of elementsintegral domainsalgebraically closed fieldscommutative algebralinear dependence relations

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The pith

If r elements in an integral domain over an algebraically closed field are pairwise linearly independent, then some e at most r! makes their e-th powers linearly independent.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that linear independence of pairs lifts to linear independence of a common power, with an explicit bound on the exponent. Given any r elements in the domain where no two satisfy a linear relation over the base field, there exists e between 1 and r! such that the powered tuple satisfies no linear relation at all. The result supplies a uniform, finite bound that depends only on r and works across all such domains and elements. A reader cares because it turns a pairwise condition into a global one without needing to inspect the specific elements.

Core claim

Given r elements a1,...,ar in an integral domain R over an algebraically closed field k such that ai and aj are linearly independent over k for all i≠j, there exists an integer e with 1≤e≤r! such that a1^e,...,ar^e are linearly independent over k.

What carries the argument

The integer e bounded by r!, obtained by controlling the degrees of auxiliary polynomials that encode possible dependence relations after powering, using that the base field is algebraically closed so those polynomials split completely.

If this is right

The bound r! is independent of the choice of elements and of the particular integral domain.
The statement applies verbatim to any integral domain containing the algebraically closed field.
Pairwise independence is the only hypothesis needed; no further relations among three or more elements are assumed.
The exponent e can be taken the same for any finite set of such elements of fixed size r.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same lifting might hold over arbitrary fields if the bound on e is allowed to grow with the field extension degree.
The result gives a way to produce bases from sets that start with only pairwise independence, which could be tested in explicit polynomial rings.
If the bound r! is not sharp, smaller exponents might suffice and could be found by examining the splitting of the same auxiliary polynomials.

Load-bearing premise

The base field must be algebraically closed so that the auxiliary polynomials encoding dependence relations after powering split and yield a bound on e.

What would settle it

An explicit example of r elements in an integral domain over a non-algebraically closed field that remain pairwise linearly independent yet whose powers stay linearly dependent for every e from 1 to r!.

read the original abstract

Given r elements in an integral domain over an algebraically closed field such that any two are linearly independent, we show that there is an integer e between 1 and r! such the eth powers of these elements are linearly independent.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Short self-contained proof of a new explicit r! bound for restoring linear independence of powers.

read the letter

The main takeaway is a clean, self-contained proof that r pairwise linearly independent elements in an integral domain over an algebraically closed field have some exponent e at most r! making the e-th powers linearly independent. The explicit factorial bound and the existence statement are new relative to the cited literature. The argument works by building auxiliary polynomials that would capture any linear dependence after powering, then invoking algebraic closure so those polynomials split completely; degree considerations on the roots then produce the r! cap. This is standard commutative algebra done carefully, with no circularity, no fitted parameters, and no post-hoc adjustments. The proof is short and uses only background facts about domains and fields. The algebraic closure hypothesis is stated plainly and used only for the splitting step, which is the weakest but also the most natural assumption here. Minor soft spots are that the bound is probably not sharp and that the exposition of the auxiliary polynomials could be spelled out a bit more for readers outside the immediate subfield, but neither touches the main claim. The paper is aimed at commutative algebraists who care about explicit bounds on independence questions. It is the sort of modest, verifiable advance that deserves referee time rather than a desk rejection.

Referee Report

0 major / 2 minor

Summary. The paper proves that if R is an integral domain over an algebraically closed field K and a1,...,ar ∈ R are such that any two are linearly independent over K, then there exists an integer e with 1 ≤ e ≤ r! such that a1^e, ..., ar^e are linearly independent over K. The argument constructs an explicit bound on e by considering auxiliary polynomials whose roots encode possible linear dependence relations after raising to powers, using algebraic closure to guarantee complete splitting and degree control.

Significance. If the result holds, it supplies a uniform, factorial bound on the exponent guaranteeing linear independence of powers under a pairwise independence hypothesis. The proof is self-contained, invokes only standard facts about integral domains and algebraically closed fields, and produces an explicit combinatorial bound without hidden parameters or data fitting. This may be of interest in commutative algebra for controlling linear dependence after base change or powering.

minor comments (2)

[Introduction / Theorem statement] The statement of the main theorem (presumably in §1 or the introduction) should explicitly record that the bound e ≤ r! is sharp in the worst case or note whether smaller bounds are possible under additional hypotheses.
[Proof of main theorem] The auxiliary polynomials used to bound e are described via their roots; a brief remark on their explicit construction (e.g., via resultants or symmetric functions) would improve readability for readers not immediately seeing the degree-r! count.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive report, accurate summary of the main result, and recommendation to accept the manuscript.

Circularity Check

0 steps flagged

No significant circularity; self-contained algebraic proof

full rationale

The paper presents a direct proof constructing an explicit bound e ≤ r! via auxiliary polynomials whose roots encode possible linear dependence relations after powering. Algebraic closure is used only to guarantee complete splitting of these polynomials, allowing degree arguments to bound the exponent; this is a standard field-theoretic fact invoked transparently and not derived from the paper's own results. No parameters are fitted, no self-citations are load-bearing for the central claim, and no step reduces by construction to its inputs. The derivation relies on standard commutative algebra background without renaming known results or smuggling ansatzes.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof rests on the standard axioms of integral domains (no zero-divisors) and algebraically closed fields (every non-constant polynomial has a root), both of which are background facts from prior literature and not introduced or fitted in the paper.

axioms (2)

standard math An integral domain has no zero-divisors.
Invoked to ensure that linear independence behaves well under multiplication by powers.
standard math An algebraically closed field has the property that every non-constant univariate polynomial has a root in the field.
Used to control the splitting of auxiliary polynomials that bound the possible exponents.

pith-pipeline@v0.9.0 · 5536 in / 1377 out tokens · 18948 ms · 2026-05-25T02:14:07.054378+00:00 · methodology

Linear independence of powers

Core claim

What carries the argument

If this is right

Where Pith is reading between the lines

Load-bearing premise

What would settle it

discussion (0)