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This paper proves that for s homogeneous degree-d forms in n variables, resultants of generic linear combinations, taken for a generic set of binomial(d+n-1,n-1)*s - n^2 + 1 matrices, form a resultant system; this improves all known upper b

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2026-08-03 09:32 UTC pith:GSJNJRQS

load-bearing objection Real advance in elimination theory: resultant systems of polynomial size for fixed n, with bivariate O(sd) matching Lyubeznik's lower bound up to a constant; incidence-count proof is coherent, though it leans on Perron's theorem as a black box.

arxiv 2607.29308 v1 pith:GSJNJRQS submitted 2026-07-31 math.AC cs.SCmath.AG

Small Resultant Systems via Linear Combinations

M. Levent Do\u{g}an , Elias Tsigaridas , Zafeirakis Zafeirakopoulos This is my paper
classification math.AC cs.SCmath.AG MSC 13P1514L24
keywords resultant systemMacaulay resultantelimination theoryGIT quotientdeterminantal varietylinear combinationsSylvester resultantprojective variety
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper's central claim is that a resultant system — a finite set of polynomial conditions on the coefficients that detects a common nonzero solution — can be built from far fewer polynomials than earlier constructions. For s homogeneous degree-d forms in n variables, the authors prove that taking the classical Macaulay resultant of n generic linear combinations of the forms, for a generic set of binomial(d+n-1,n-1)*s - n^2 + 1 coefficient matrices, yields such a system. This count is Theta(s d^(n-1)) when n is fixed, which is polynomial rather than exponential in d or s. For bivariate systems (n=2) the construction gives (d+1)s - 3 polynomials, matching a known lower bound up to a constant factor. The authors also give fully explicit 'punctured' resultant systems, assuming every input form is nonzero, with (s-2)d+1 polynomials in the bivariate case and poly(s,d) in general for fixed n.

Core claim

On the paper's own terms, the central discovery is that the resultant variety — the irreducible locus of coefficient tuples admitting a common projective zero — can be cut out set-theoretically by resultants of linear combinations, evaluated at a generic set of matrices whose size is exactly the dimension of the projective GIT quotient of the coefficient space by SL(n), plus one. Theorem 1.5 states that if H is a generic set of n×s matrices with |H| ≥ binomial(d+n-1,n-1)*s - n^2 + 1, then the collection {Res(Λ11f1+...+Λ1sfs, ..., Λn1f1+...+Λnsfs) : Λ in H} is a resultant system. The proof shows that for any f with no common zero, the set of matrices Λ for which this resultant vanishes has co

What carries the argument

The load-bearing object is the projective GIT quotient of P(K[x]_d^s) by the natural SL(n)-action via change of variables; its dimension equals binomial(d+n-1,n-1)*s - n^2, so one extra matrix yields the claimed count. Two auxiliary results carry the argument: Perron's theorem, which guarantees that a generic n×s matrix of linear combinations preserves whether the system has a common nonzero solution, and the incidence-variety dimension count that converts that codimension-one property into a genericity statement about a finite collection of matrices.

Load-bearing premise

The proof leans on Perron's theorem, cited from 1941 without proof: a generic n×s matrix of linear combinations preserves whether the input forms have a common nonzero solution; if this theorem were false, the codimension-one step that drives the entire dimension count would collapse.

What would settle it

Compute the three Sylvester resultants from Example 4.3 — Res(f1,f3), Res(f1+f2,f3), Res(f1−f2,f3) — for a randomly chosen triple of nonzero binary quadratics with no common projective root; if all three vanish for such a triple, the explicit punctured construction in Theorem 1.8 fails.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • For fixed n, resultant systems of cardinality Theta(s d^(n-1)) exist — polynomial in both s and d, and the first such bounds for n>2.
  • For bivariate systems of equal degree d, a resultant system of (d+1)s-3 polynomials exists, closing the gap between earlier O(s^2 d) and O(s d^2) upper bounds and the known Omega(s d) lower bound up to a constant factor.
  • For linear systems (d=1), generic collections of sn-n^2+1 matrices yield determinants det(Λ_i C) that define the determinantal variety of rank-deficient s×n matrices set-theoretically.
  • Assuming every input form is nonzero, explicit punctured resultant systems exist: (s-2)d+1 Sylvester resultants in the bivariate case, and O(s^(n-1) d^(n(n-1)/2)) polynomials for fixed n>2.
  • These punctured systems use arbitrary distinct scalars, not generic choices, so they can be written down immediately.

Where Pith is reading between the lines

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

  • The cardinality being the GIT quotient dimension plus one suggests a general principle: any homogeneous SL(n)-equivariant hypersurface condition on the coefficient space might admit a resultant-system description with that many evaluations; a testable extension would be to replace the Macaulay resultant by other invariant hypersurfaces.
  • The linear-case Corollary 1.6 offers a potentially cheap membership test for the determinantal variety: if a fixed generic tuple (Λ_1,...,Λ_N) is precomputed, checking whether a given matrix C has rank < n reduces to evaluating N determinants, avoiding all maximal minors — though the paper leaves finding explicit such tuples open.
  • The punctured construction, built on Vandermonde linear combinations and a pigeonhole argument, suggests algorithms where systems are certified by evaluating only O(sd) resultants after fixing scalar parameters; it would be interesting to see whether the non-punctured generic bound can be matched with explicit matrices for n>2.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 5 minor

Summary. The paper studies resultant systems for s homogeneous polynomials of common degree d in n variables. A resultant system is a set of polynomials in the coefficient space whose simultaneous vanishing characterizes the existence of a common nonzero projective solution. The main result (Theorem 1.5) states that for a generic collection H of n×s matrices with |H| ≥ N := binom(d+n−1,n−1)s − n² + 1, the set {Res(Λ·f) : Λ∈H} is a resultant system. The proof is via an incidence-variety dimension count over a projective GIT quotient of the space of s-tuples by SL(n), combined with Perron's theorem on linear combinations preserving the (non)emptiness of the common zero locus. The paper also gives a self-contained treatment of the linear case (Corollary 1.6), recovering the Bruns–Schwänzl cardinality with a new proof, and constructs explicit 'punctured' resultant systems under the assumption that no input polynomial is zero: (s−2)d+1 polynomials in the bivariate case and O(s^{n−1}d^{n(n−1)/2}) polynomials for fixed n>2 (Theorems 1.8 and 4.5).

Significance. If correct, the main theorem is a substantial improvement over previous resultant-system constructions: the cardinality is Θ(s d^{n−1}) for fixed n, whereas classical constructions (Kapferer, Abramov, Yap, Orsinger–Kakié) are larger and, in several regimes, exponential or super-polynomial. The linear case gives a new, compact set-theoretic description of determinantal varieties. The punctured constructions are fully explicit and do not rely on generic choices. The proof strategy—using the dimension of the GIT quotient to control the number of linear combinations needed—is elegant and likely to be useful beyond this specific problem. The main external dependence is Perron's theorem (Theorem 2.8), but this is a classical, published result and the paper cites both the original source and modern references. The manuscript is clearly written and the technical steps are reproducible.

minor comments (5)
  1. [§3.3] After defining V = φ(U), the text says 'V=X since X is irreducible.' This is false: irreducibility only implies V is dense in X. The subsequent argument uses only dim V = dim X, so the proof is unaffected, but the sentence should be corrected to 'V is dense in X'.
  2. [§2.4, Theorem 2.8] Perron's theorem is cited from [Per41] and used as a black box in two load-bearing places: the codimension statement (3.1) and Corollary 3.8 (semistability of no-solution systems). While the theorem is standard and the citations are appropriate, a short proof sketch or a more precise statement of the generic-Λ version would make the paper more self-contained and help readers verify the exact hypotheses (equal degree, char 0, s≥n). This is a presentation suggestion, not a correctness concern.
  3. [§4.2, Theorem 4.5] The phrase 'the cardinality of C_{n−2} is at most ≤ d^{n−1}' contains a duplicated comparison ('at most ≤'). Also, the notation f_{λ_i} in the statement is defined only after the display; consider defining it before the display for readability.
  4. [§1.1.1] The statement that Kapferer's construction yields a number of polynomials 'exponential in d and/or s' is imprecise: for fixed s the count is polynomial in d, and for fixed d it is polynomial in s. It is super-polynomial only in joint regimes such as s ~ d. The later comparison with the new O(sd) bound is still valid, but the wording could be sharpened.
  5. [§3.2, proof of Theorem 3.5] The notation 'SL[f]' is used for the stabilizer of [f]; this is nonstandard and could be confused with the image of f under SL. Write instead Stab_{SL(n)}([f]) or define the notation explicitly.

Circularity Check

0 steps flagged

No significant circularity; the derivation is self-contained modulo standard external theorems.

full rationale

The paper's main construction is not circular. Theorem 1.5's resultant system consists of Macaulay resultants evaluated at a generic set of matrices of size N = (d+n-1 choose n-1)s - n^2 + 1. This number is obtained from a genuine incidence-variety dimension count over the GIT quotient, not by fitting the target vanishing locus; no parameter is selected after inspecting the f. The most load-bearing input, Perron's theorem (Theorem 2.8), is cited as an external classical result ([Per41, CG83, Koi00]) and used to guarantee that for any system with no common projective zero there is a linear combination whose square resultant is nonzero. That yields codimension 1 for the pullback of the square-resultant hypersurface and supplies the invariant needed for semistability. This is legitimate black-box use of an external theorem, not a self-citation or an ansatz smuggled in by the same authors. The GIT quotient properties in Theorem 3.5 are proved in the paper from standard sources (Hoskins, Mumford-Fogarty-Kirwan, Luna), and no uniqueness theorem of the authors is invoked to force the choice. The punctured resultant systems in Section 4 use Vandermonde invertibility, Bezout bounds, and pigeonhole arguments; they do not rename a known result or fit a prediction. The only notable slip is the sentence in Section 3.3, 'Moreover, V=X since X is irreducible,' which overstates an open dense subset as equal to the whole irreducible variety; however, the subsequent dimension computation only uses dim V = dim X, so the slip is not load-bearing and is not a circularity. Overall, no step reduces the claimed prediction to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

No free parameters, no fitted constants, and no invented entities. The central claim rests on standard external theorems in commutative algebra and geometric invariant theory; the only domain-specific assumption is the algebraically closed characteristic-0 field.

axioms (5)
  • standard math Perron's theorem (Thm 2.8): a generic n×s matrix of linear combinations preserves the (non-)emptiness of the common zero set
    Used in Theorem 1.4 proof, Corollary 3.8, and in (3.1) to obtain a codimension-one pullback; cited from [Per41].
  • standard math Macaulay's resultant is SL(n)-invariant and vanishes exactly when an n-tuple of forms has a common projective zero
    Invoked in Lemma 3.7 and throughout the incidence-counting proof; cited to [CLO05] and [Jou91].
  • standard math Hilbert-Nagata finite generation and the GIT quotient construction for reductive SL(n) actions
    Theorem 3.5 uses these to define the quotient and its dimension; cited to [Hos12] and [MFK94].
  • standard math Luna's theorem: finite generic stabilizer implies closed generic orbit
    Corollary 3.11 uses it to show the generic fiber dimension of the quotient is n^2-1; cited to [Lun73] and [Kra84].
  • domain assumption K is algebraically closed of characteristic 0
    Declared at the start; used for resultant properties, Perron's theorem, and GIT results.

pith-pipeline@v1.3.0-daily-deepseek · 24062 in / 28562 out tokens · 273524 ms · 2026-08-03T09:32:10.356114+00:00 · methodology

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read the original abstract

For a system of $s$ homogeneous polynomials of degree $d$ in $n$ variables, say ${\bf{f}} = 0$, we consider the problem of constructing resultant systems. A resultant system is a finite set of polynomials in the coefficients of the input polynomials, the vanishing of which characterizes the systems $\bf{f}$ with a common non-zero solution. The classical approaches for constructing resultant systems rely either on maximal minors of large coefficient matrices or on the coefficients of a resultant of generic linear combinations of the input polynomials. Typically, they produce resultant systems containing a very large number of polynomials. We develop new constructions based on taking resultants of linear combinations of the input polynomials; this results in resultant systems of small cardinality. Our main results are: 1) We prove that a resultant system with ${d+n-1 \choose n-1} s-n^2+1$ polynomials exists; each polynomial is the resultant of $n$ linear combinations of the input polynomials. This improves the previously known upper bounds, even for systems of bivariate homogeneous polynomials. 2) Under the assumption that the input polynomials are non-zero, we construct explicit resultant systems with cardinality $\mathrm{poly}(s,d)$, when $n$ is fixed.

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