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The symmetric determinantal complexity of sum x_i^n is at least (1/(2e) - o(1)) n^2 over the complex numbers.

2026-06-27 10:50 UTC pith:VP7MR5Q3

load-bearing objection This paper gives a new quadratic lower bound on symmetric determinantal complexity for diagonal power sums by adapting polar-degree incidence geometry to the symmetric case. the 2 major comments →

arxiv 2606.11090 v1 pith:VP7MR5Q3 submitted 2026-06-09 cs.CC math.AG

A symmetric determinantal lower bound for diagonal power sums via polar degree

Karthik Sheshadri This is my paper
classification cs.CC math.AG
keywords symmetric determinantal complexitypolar degreediagonal power sumslower boundshypersurfaceSchur complementincidence varietydeterminant
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 proves a lower bound on the size of the smallest symmetric matrix of linear forms whose determinant equals a given polynomial. For the sum of nth powers in n variables the matrix must have size at least roughly n squared divided by 2e. The argument connects this size to the top polar degree of the hypersurface defined by the polynomial through an incidence analysis that uses a symmetric Schur-complement normal form at corank-one points. A reader would care because the bound is concrete, applies to a natural family of polynomials, and is accompanied by an explicit construction showing it is tight up to a constant factor.

Core claim

If a smooth degree-d hypersurface X = V(f) in P^{N-1} admits a symmetric determinantal representation of size m, then its top polar degree d(d-1)^{N-2} is at most 2^{N-2} binom(m, N-1). Specializing to f equal to the sum of dth powers of N variables produces the lower bound sdc(F_{N,d}) >= (1/(2e) - o_N(1)) N(d-1) as N tends to infinity, and the case d = n, N = n recovers the quadratic bound on sdc(sum x_i^n).

What carries the argument

The symmetric rank-one kernel incidence M(z,x) u = 0 together with the symmetric Schur-complement normal form that eliminates the kernel line scheme-theoretically and aligns the lifted conormal forms with the partial derivatives of f.

Load-bearing premise

The hypersurface defined by the polynomial is smooth.

What would settle it

An explicit symmetric matrix of size smaller than (1/(2e) - epsilon) n^2 whose determinant equals sum x_i^n for arbitrarily large n would falsify the lower bound.

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

If this is right

  • Any smooth hypersurface of degree d in P^{N-1} obeys the polar-degree upper bound 2^{N-2} binom(m, N-1) on its top polar degree.
  • The diagonal forms F_{N,d} require symmetric determinantal size at least roughly N(d-1)/(2e) for large N.
  • An explicit symmetric representation of F_{N,d} of size 2N(d+1)+1 exists, so the lower bound is tight up to a constant factor.
  • The stated bounds hold only for exact complexity in characteristic zero.

Where Pith is reading between the lines

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

  • The same polar-degree technique could be tested on other families of polynomials whose hypersurfaces are smooth.
  • Computational search for representations of small diagonal forms might reveal whether the constant 1/(2e) can be improved.
  • The o_N(1) term vanishing as N grows suggests the leading coefficient becomes asymptotically sharp for the general diagonal case.

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

2 major / 2 minor

Summary. The manuscript proves that the symmetric determinantal complexity sdc(sum_{i=1}^n x_i^n) is at least (1/(2e) - o(1)) n^2 over the complex numbers. More generally, for the diagonal form F_{N,d} = sum_{i=1}^N x_i^d with N >= 3, it shows sdc(F_{N,d}) >= (1/(2e) - o_N(1)) N(d-1) as N -> infinity. The argument proceeds by showing that if a smooth degree-d hypersurface X = V(f) in P^{N-1} admits a symmetric determinantal representation of size m, then its top polar degree d(d-1)^{N-2} is bounded above by 2^{N-2} C(m, N-1) via an incidence variety M(z,x) u = 0, symmetric Schur-complement elimination of the kernel line at genuine polar points, and multihomogeneous Bézout on P^N x P^{m-1}. An explicit symmetric representation of size 2N(d+1)+1 is supplied to show the bounds are non-vacuous.

Significance. If the local incidence analysis holds, the result supplies a parameter-free geometric lower-bound technique for exact symmetric determinantal complexity that produces explicit asymptotic constants from polar degrees and Bézout numbers. It is a self-contained symmetric companion to the author's earlier non-symmetric work and includes a matching upper-bound construction, making the constants tight up to a fixed factor. This strengthens the toolkit for algebraic complexity lower bounds in the symmetric setting.

major comments (2)
  1. [local incidence analysis with symmetric Schur-complement normal form] The local incidence analysis (proof of the general theorem): the assertion that, at a genuine polar point where rank(M) = m-1, the symmetric Schur-complement normal form eliminates the unique kernel line scheme-theoretically, so that the lifted conormal forms u^T A_i u become a common non-vanishing unit multiple of the partials d_i f, is load-bearing for the subsequent claim that the lifted polar equations cut the ordinary polar slice up to units and that each genuine lifted polar point is an isolated zero-dimensional solution. Explicit local coordinates or equations verifying that the multiplier is a unit in the local ring and that no extra components arise are required; otherwise the inequality d(d-1)^{N-2} <= 2^{N-2} C(m, N-1) does not follow.
  2. [application to diagonal power sums] Application to F_n and the asymptotic (section deriving the constant 1/(2e)): the o(1) term in (1/(2e) - o(1)) n^2 and the general (1/(2e) - o_N(1)) N(d-1) must be traced explicitly to the ratio of the polar degree d(d-1)^{N-2} to the Bézout number 2^{N-2} C(m, N-1) when m is taken minimal; any hidden dependence on N or d in the constant C(m, N-1) would affect the claimed limit.
minor comments (2)
  1. Define the multihomogeneous Bézout number C(m, N-1) explicitly (e.g., as the coefficient of the appropriate monomial in the product of the bi-homogeneous equations) rather than leaving it implicit.
  2. [introduction] The relation between the current symmetric argument and the cited non-symmetric preprint should be stated more precisely in the introduction, highlighting exactly which parts of the incidence analysis are redone.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the significance of the symmetric polar-degree technique. We address the two major comments below. We agree that the local incidence analysis would benefit from explicit coordinate charts to make the scheme-theoretic elimination and unit multiplier fully transparent, and we will add them. We will also expand the asymptotic section to trace the o(1) term explicitly from the ratio of polar degree to the multihomogeneous Bézout number.

read point-by-point responses

  1. Referee: [local incidence analysis with symmetric Schur-complement normal form] The local incidence analysis (proof of the general theorem): the assertion that, at a genuine polar point where rank(M) = m-1, the symmetric Schur-complement normal form eliminates the unique kernel line scheme-theoretically, so that the lifted conormal forms u^T A_i u become a common non-vanishing unit multiple of the partials d_i f, is load-bearing... Explicit local coordinates or equations verifying that the multiplier is a unit in the local ring and that no extra components arise are required.

    Authors: We agree that the current write-up of the symmetric Schur-complement step, while correct in outline, would be strengthened by an explicit local coordinate chart. In the revision we will add a dedicated paragraph (immediately after the definition of the incidence variety M(z,x)u=0) that works in affine coordinates centered at a genuine polar point p where the kernel line is spanned by the last standard basis vector e_m. We exhibit the block form of the symmetric matrix after elementary row/column operations that preserve symmetry, compute the Schur complement explicitly, and verify that the resulting multiplier on u^T A_i u is a unit in the local ring (its constant term is nonzero because the point is genuine). We also confirm that the ideal generated by the lifted equations is radical and zero-dimensional on the polar slice, with no embedded components, by direct computation of the Jacobian criterion in these coordinates. This makes the passage from the incidence equations to the bound d(d-1)^{N-2} ≤ 2^{N-2} C(m,N-1) fully rigorous. revision: yes

  2. Referee: [application to diagonal power sums] Application to F_n and the asymptotic (section deriving the constant 1/(2e)): the o(1) term in (1/(2e) - o(1)) n^2 and the general (1/(2e) - o_N(1)) N(d-1) must be traced explicitly to the ratio of the polar degree d(d-1)^{N-2} to the Bézout number 2^{N-2} C(m, N-1) when m is taken minimal; any hidden dependence on N or d in the constant C(m, N-1) would affect the claimed limit.

    Authors: We will add a new subsection (immediately after the statement of the general theorem) that computes the asymptotic explicitly. Let m be the smallest integer satisfying d(d-1)^{N-2} ≤ 2^{N-2} C(m,N-1). The multihomogeneous Bézout number C(m,N-1) is a polynomial in m of degree N-1 whose leading coefficient is independent of d and whose lower-order terms contribute only o(1) when N o∞ with d fixed or d growing slower than any positive power of N. Taking (N-1)-th roots and applying Stirling’s approximation to the resulting binomial coefficients yields lim (polar degree / Bézout number)^{1/(N-1)} = 1/(2e) uniformly in the stated regime; the o_N(1) term is therefore precisely the contribution of the lower-degree terms in C(m,N-1) and vanishes as N o∞. The same calculation produces the claimed (1/(2e)-o(1))n^2 bound for the diagonal power sum when N=n and d=n. revision: yes

Circularity Check

0 steps flagged

No circularity: bound derived from independent geometric incidence count

full rationale

The central inequality sdc(f) >= polar degree / (2^{N-2} C(m,N-1)) follows from the top polar degree d(d-1)^{N-2} of a smooth hypersurface together with a scheme-theoretic local incidence analysis (symmetric Schur complement eliminating the kernel line at genuine polar points, lifted conormal forms being unit multiples of partials, and multihomogeneous Bézout on P^N x P^{m-1}). This argument is presented as self-contained in the symmetric setting; the cited non-symmetric preprint is used only for comparison and is not load-bearing. No fitted parameters, self-definitional relations, or reductions of the claimed constants to prior fitted values occur. The explicit symmetric representation of F_{N,d} is given separately and does not enter the lower-bound derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard algebraic-geometry facts (multihomogeneous Bezout, polar degree formula for smooth hypersurfaces) plus the domain assumption that the hypersurface is smooth; no free parameters or invented entities are introduced.

axioms (2)
  • domain assumption X = V(f) in P^{N-1} is a smooth degree-d hypersurface with N >= 3
    Invoked explicitly in the general theorem to guarantee the polar degree equals d(d-1)^{N-2} and to justify the local incidence analysis.
  • standard math Multihomogeneous Bezout theorem applies to the incidence variety in P^N x P^{m-1}
    Used to obtain the bound 2^{N-2} C(m, N-1) on the number of genuine lifted polar points.

pith-pipeline@v0.9.1-grok · 5982 in / 1550 out tokens · 29529 ms · 2026-06-27T10:50:42.027057+00:00 · methodology

0 comments
read the original abstract

The symmetric determinantal complexity sdc(f) of a polynomial f is the least m such that f = det(M) for an m x m symmetric matrix M of affine-linear forms. We prove, over the complex numbers, that sdc(sum_{i=1}^n x_i^n) >= (1/(2e) - o(1)) n^2. This is a symmetric companion to the author's non-symmetric polar-degree preprint (arXiv:7680505); the method parallels that work, but the proof here is self-contained and redoes the load-bearing local incidence analysis in the symmetric setting. The general theorem: if X = V(f) in P^{N-1} is a smooth degree-d hypersurface, N >= 3, and f = det(A_0 + sum x_i A_i) with all A_i symmetric of size m, then the top polar degree d(d-1)^{N-2} is at most 2^{N-2} C(m, N-1). The proof uses the symmetric rank-one kernel incidence M(z,x) u = 0. At a genuine polar point M has rank m-1, and a symmetric Schur-complement normal form eliminates the unique kernel line scheme-theoretically; on the resulting local graph the lifted conormal forms u^T A_i u are a common unit multiple of the partials d_i f, so the lifted polar equations cut the ordinary polar slice up to units and each genuine lifted polar point is a zero-dimensional isolated solution. Multihomogeneous Bezout on P^N x P^{m-1} then yields the bound 2^{N-2} C(m, N-1). For F_n = sum x_i^n this gives the constant 1/(2e). More generally, for F_{N,d} = sum_{i=1}^N x_i^d the same theorem gives sdc(F_{N,d}) >= (1/(2e) - o_N(1)) N(d-1) as N -> infinity. We give an explicit symmetric representation of F_{N,d} of size 2N(d+1)+1, so the diagonal bounds are non-vacuous and tight up to a constant. The result is for exact symmetric determinantal complexity in characteristic zero; it is not a border statement and not a uniform positive-characteristic theorem.

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Reference graph

Works this paper leans on

20 extracted references · 2 canonical work pages · 1 internal anchor

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    SYMMETRIC ISOLATEDNESS. Does each genuine polar point lift to a ZERO-DIMENSIONAL, scheme-theoretically isolated solution of the symmetric incidence system {M(x)u=0, q_j(u^T A_i u)=0, slice}? Prove it via the symmetric local normal form: with M = [[B,c],[c^T,s]], det B a unit near the point, kernel u=(-B^{-1}c, 1), and det M = (det B)(s - c^T B^{-1} c). Ve...

  12. [12]

    Confirm q_j(u^T A_i u) is genuinely class 2U and that the symmetric incidence is a SQUARE system on P^N x P^{m-1} (count equations vs dimensions)

    THE MULTIDEGREE 2U. Confirm q_j(u^T A_i u) is genuinely class 2U and that the symmetric incidence is a SQUARE system on P^N x P^{m-1} (count equations vs dimensions). Verify no analogue of the redundant-left-equation reduction is silently needed or silently missing

  13. [13]

    Verify [H^N U^{m-1}] H (H+U)^m (2U)^{N-2} = 2^{N-2} C(m, N-1) exactly, and that the root extraction yields the constant 1/(2e), not 1/(4e) or something else

    THE BEZOUT EXTRACTION. Verify [H^N U^{m-1}] H (H+U)^m (2U)^{N-2} = 2^{N-2} C(m, N-1) exactly, and that the root extraction yields the constant 1/(2e), not 1/(4e) or something else

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    EXISTENCE / NONDEGENERACY. Does a smooth symmetric determinantal representation of F_n even exist for the relevant m, and is the generic polar section reduced of the right cardinality in the symmetric setting? If symmetric representations of F_n are obstructed, the theorem is vacuous or false -- check this. Close with: (i) overall verdict on the n^2 bound...

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    Rank-drop escape cannot pass through a genuine point because the local normal form is taken on the open set z̸ = 0 and detB̸ = 0, and the local incidence is a graph over g = 0; see Lemma 3

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    A positive-dimensional spurious component cannot pass through a genuine point because on the local graph the lifted polar equations are ordinary polar equations multiplied by units; see (6)

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    Nonreduced isolated multiplicity is allowed and is exactly what Bezout counts

    Nonreduced structure is controlled because the completed local incidence ring is Artinian. Nonreduced isolated multiplicity is allowed and is exactly what Bezout counts

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    No left-kernel reduction is present in the symmetric system

    The polar equations have class 2 U because uTAiu is a quadratic form in the single projective kernel variable. No left-kernel reduction is present in the symmetric system

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    The coefficient extraction is the elementary identity [H N U m−1]H(H+U) m(2U) N−2 = 2N−2 m N−1

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    The model is nonempty forF n by Proposition 1. 14