{"id":"2cd5bd93-9655-45e3-a138-47b89ed66aeb","arxiv_id":"2606.11090","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Proves sdc(sum x_i^n) >= (1/(2e) - o(1)) n^2 over C using polar degree of the associated hypersurface and multihomogeneous Bezout on an incidence variety after symmetric Schur complement.","lead":"The paper proves that the symmetric determinantal complexity of the polynomial sum of nth powers is at least roughly n squared over 2e. A smart generalist might read it to see how algebraic geometry tools produce concrete lower bounds on matrix representations of polynomials.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Correctness of symmetric Schur-complement elimination of kernel line at genuine polar points","rationale":"The reader's weakest assumption is precisely the load-bearing local incidence step. The abstract states the proof is self-contained and redoes this analysis symmetrically, but the correctness of that step remains the single point on which the entire quantitative bound depends; without independent verification of the scheme-theoretic claims, the result stays unverified. No other internal inconsistency is visible from the given statement.","tokens_in":2009,"tokens_out":451,"duration_ms":23617,"concrete_test":"Extract the local coordinate chart and normal-form calculation for the symmetric incidence M(z,x)u=0 (the section redoing the incidence analysis); substitute a generic genuine polar point into the Schur-complement matrix and verify that the kernel line is cut out scheme-theoretically with multiplicity one and that the common multiplier relating u^T A_i u to d_i f is non-zero; count the resulting isolated points against the polar degree.","verdict_should_be":"UNVERDICTED","load_bearing_attack":"The lower bound sdc(f) >= polar degree / (2^{N-2} C(m,N-1)) for smooth degree-d hypersurface f in P^{N-1} rests entirely on the local incidence claim: at a genuine polar point where rank(M)=m-1, the symmetric Schur-complement normal form eliminates the unique kernel line scheme-theoretically; the resulting local graph makes the lifted conormal forms u^T A_i u a common non-vanishing unit multiple of the partials d_i f; consequently the lifted polar equations cut the ordinary polar slice up to units and each genuine lifted polar point is a zero-dimensional isolated solution to which multihomogeneous Bézout on P^N × P^{m-1} applies. If the scheme-theoretic elimination fails, the unit multiple vanishes at some points, or the solutions are not isolated, the inequality d(d-1)^{N-2} <= 2^{N-2} C(m,N-1) does not follow and the claimed constants 1/(2e) and (1/(2e)-o_N(1)) are unsupported.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","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.","tokens_in":2276,"tokens_out":823,"duration_ms":27111,"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":[{"comment":"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.","section":"local incidence analysis with symmetric Schur-complement normal form"},{"comment":"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.","section":"application to diagonal power sums"}],"minor_comments":[{"comment":"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.","section":null},{"comment":"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.","section":"introduction"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"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.","responses":[{"response":"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_made":"yes","referee_comment":"[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."},{"response":"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\to∞ 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\to∞. 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_made":"yes","referee_comment":"[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."}],"tokens_in":1956,"tokens_out":860,"duration_ms":19279,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The headline result is sdc of sum x_i^n is at least (1/(2e) - o(1)) n^2 over the complexes, with a more general statement for F_{N,d} giving roughly N(d-1)/(2e) as N grows. This is a direct symmetric companion to the author's earlier non-symmetric preprint.\n\nThe paper does the work of redoing the local incidence analysis in the symmetric setting. It states a general bound: for a smooth degree-d hypersurface in P^{N-1}, the top polar degree is at most 2^{N-2} times the binomial C(m, N-1). The argument lifts the polar equations via the symmetric rank-one kernel incidence, applies a symmetric Schur-complement normal form to remove the kernel line scheme-theoretically at genuine polar points, and then invokes multihomogeneous Bezout on the product space. They also supply an explicit symmetric representation of size 2N(d+1)+1, so the lower bound is not vacuous.\n\nThe central technical step is that Schur-complement elimination and the claim that the lifted conormal forms are unit multiples of the partials. The abstract presents this as completed and self-contained, so the bound stands if those local details hold without extra components or vanishing factors. No other gaps are visible from the given material.\n\nThis is for people working on algebraic complexity and geometric lower bounds. A reader who follows determinantal representations or polar methods will get a concrete new constant and a reusable general theorem. It deserves a serious referee because the result is new, the method is explicit, and the upper-bound construction makes the comparison meaningful.","headline":"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.","tokens_in":2772,"tokens_out":414,"would_cite":false,"duration_ms":19681,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"The symmetric determinantal complexity of sum x_i^n is at least (1/(2e) - o(1)) n^2 over the complex numbers.","keywords":["symmetric determinantal complexity","polar degree","diagonal power sums","lower bounds","hypersurface","Schur complement","incidence variety","determinant"],"falsifier":"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.","tokens_in":2904,"feed_emoji":"","tokens_out":762,"duration_ms":29270,"temperature":0.7,"pith_summary":"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.","feed_headline":"Symmetric det size for sum of nth powers at least n²/(2e)","feed_subtitle":"Polar-degree incidence argument shows matrix size must grow quadratically for exact representation of the diagonal form.","key_machinery":"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.","core_discovery":"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).","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Lower bound n²/(2e) for symmetric determinantal complexity of power sums","Polar incidence yields n²/(2e) lower bound on symmetric det size","Symmetric det size lower bound n²/(2e) for diagonal power sums","Quadratic lower bound n²/(2e) on symmetric det complexity for powers"],"cache_read_input_tokens":64,"weakest_assumption_plain":"The hypersurface defined by the polynomial is smooth.","fun_headline_variants_meta":{"raw":{"variants":["Lower bound n²/(2e) for symmetric determinantal complexity of power sums","Polar incidence yields n²/(2e) lower bound on symmetric det size","Symmetric det size lower bound n²/(2e) for diagonal power sums","Quadratic lower bound n²/(2e) on symmetric det complexity for powers"]},"model":"grok-4.3","cost_usd":0.010984,"raw_usage":{"total_tokens":4983,"prompt_tokens":964,"num_sources_used":0,"completion_tokens":83,"cost_in_usd_ticks":109837000,"prompt_tokens_details":{"text_tokens":964,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3936,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":964,"tokens_out":83,"duration_ms":27992,"temperature":1.0,"reasoning_tokens":3936,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-27T10:50:42.027057+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"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.","supporting_citations":[],"review_version":1}