Nuclear correlation functions using first-principle calculations of lattice quantum chromodynamics

Arpith Kumar; Debsubhra Chakraborty; Nilmani Mathur; Piyush Srivastava

REVIEW 2 major objections 5 minor 1 cited by

Nuclear correlation functions using first-principle calculations of lattice quantum chromodynamics

T0 review · 2 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper claims that nuclear two-point correlation functions in lattice QCD can be computed much faster by first finding a propagator-independent minimal set of determinant contractions, and that an extremely small fraction of…

desk verdict Genuinely useful randomized redundancy elimination for lattice QCD contractions, but the truncation-based speedup claim needs controlled systematics before it can be believed. read the letter →

arxiv 2411.08962 v1 pith:FA5ZDSOI submitted 2024-11-13 hep-lat hep-phnucl-th

Debsubhra Chakraborty , Piyush Srivastava , Arpith Kumar , Nilmani Mathur This is my paper · ORCID

classification hep-lathep-phnucl-th PACS 12.38.Gc

keywords latticeQCDnuclearcorrelationfunctionsWickcontractionsdeterminantalgorithmrandomizedidentitytestingTensorFlowGPUaccelerationspin-colorsymmetry

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Computing nuclear correlation functions directly from lattice QCD is blocked by the Wick-contraction explosion: naive counting gives factors like $(2N_p+N_n)!(2N_n+N_p)!\,6^{2A}4^{2A}$, already $\sim 10^{73}$ for carbon. This paper claims that most of that explosion is redundant. It presents an optimized determinant algorithm that uses randomized polynomial identity testing to find, once per interpolating operator on a small lattice, a minimal set of non-redundant determinant index tuples, reducing the scaling from $O(n_u^3 n_d^3)$ to $O(n_u^3+n_d^3)$ and yielding 4x to 28x internal reductions plus an order-of-magnitude advantage over existing algorithm timings for $A=2$ to $7$. It also introduces a tensor-based programming approach that automatically generates and GPU-accelerates contraction code, which is fastest for the lighter nuclei. Along the way the paper discovers that the contributions of the minimal terms follow an extreme, sharply peaked long-tailed distribution, so that only a tiny fraction of them reconstruct the full correlator and effective mass, hinting at an unidentified spin-color symmetry. If these claims hold, first-principles lattice calculations of light nuclei up to $A\sim12$ and beyond become practical, provided the signal-to-noise problem can be controlled.

What carries the argument

The load-bearing object is the determinant form of the Wick sum, $G(t)=\sum \Theta \det M^u \det M^d \det M^s$, in which the quark permutation sum is replaced by determinants of flavor-separated propagator matrices. The preprocessing that carries the speedup is randomized identity testing: by evaluating the determinants once with a random integer propagator, equivalent index tuples are detected with an error probability bounded by $d/|C|$ per trial, yielding a minimal, propagator-independent index set. The third mechanism is the empirically observed extreme distribution of the terms, which the paper uses to argue that a tiny top fraction of determinants or blocks reconstructs the correlator and which is conjectured to reflect an unidentified spin-color symmetry.

What would settle it

Take the $^{4}$He point-sink correlator at a light quark mass on a large lattice, compute the full minimal determinant sum, and compare the effective-mass plateau with the plateau obtained from the top $0.05\%$ of terms selected on a small $24^3\times64$ lattice. If the two plateaus disagree beyond statistical error, or if another interpolating operator requires a materially different dominant set, the truncation claim fails.

Watch

Extended reading notes

Core claim

The central discovery is that the determinant representation of Wick contractions carries more symmetry than previous analyses used. Writing the two-point function as a weighted sum of products $\det M^u \det M^d \det M^s$, one finds that many of these determinants are identical up to permutation signs, either because their index tuples are related by quark permutations or because different full index choices share the same flavor sub-tuples. The paper's preprocessing step replaces the quark propagator with random integer entries, evaluates all candidate determinants once in exact integer arithmetic on a small lattice, and uses randomized polynomial identity testing to collect the unique determinant polynomials. This minimal set of index tuples is independent of the Dirac propagator, the gauge configuration, the time slice, and the lattice volume, so it is computed once per interpolating operator and then reused. The result is that the cost of Eq. (15) scales as $N N'(O(n_u^3+n_d^3))$ instead of $O(n_u^3 n_d^3)$, with measured internal reductions of 4.35, 7.30, 27.89, and 24.72 for $^{2}$H, $^{3}$He, $^{4}$He, and $^{7}$Li, and an order-of-magnitude advantage over the block algorithm in the point-sink setup. The correlator computations also reveal that the retained terms follow an extreme, sharply peaked long-tailed distribution, so that only a small fraction of them is needed to reproduce the effective mass.

Load-bearing premise

The load-bearing premise is that the small fixed fraction of the largest determinant or block terms, identified once on a small lattice at one quark mass, continues to reproduce the full correlation function at other quark masses, lattice volumes, and interpolating operators; the paper explicitly says the associated systematic uncertainties are not yet fully understood.

Editorial extensions

If this is right

Lattice QCD calculations of light nuclei up to $A\sim12$, including $^{12}$C, become computationally feasible for point and smeared sinks once the minimal set is precomputed.
GEVP correlation matrices with many interpolating operators share determinant sub-expressions, so the preprocessing can be applied across matrix entries and accelerate excited-state spectroscopy.
Three-point functions for electromagnetic and axial structure of light nuclei can be generated and GPU-accelerated through the same tensor-based code-generation pipeline.
Retaining only the dominant spin-color terms can cut correlator cost by a further order of magnitude, reducing the resources needed for multi-volume, multi-mass extrapolations.
The observed universal dominance pattern, if confirmed as a symmetry, could guide the construction of interpolating operators with better signal-to-noise for heavier nuclei.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper's heat maps show stripes where a dominant source spin-color row stays dominant across sink choices, which suggests the weight tensor $\Theta$ is nearly factorized; estimating its numerical rank across quark masses could turn the truncation from an empirical shortcut into a controlled approximation.
The same randomized preprocessing should transfer to three- and four-point functions and to form-factor matrix elements, where the contraction sums share the same determinant structure; even without truncation, the minimal set would cut those costs substantially.
If the tail statistics follow a universal extreme-value law, the fraction of terms needed for fixed accuracy may scale predictably with $A$ and quark mass, letting practitioners budget computations for nuclei beyond $A=12$ without running the full sum.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The core algorithmic contribution here is real. The randomized Schwartz-Zippel preprocessing to find a propagator-independent minimal determinant set is a clean idea with a correct probabilistic error bound, and it should be of practical use to anyone in lattice QCD doing nuclear contractions. The additive O(n_u^3 + n_d^3 + n_s^3) scaling is a straightforward tightening of Detmold-Orginos, and the paper shows it cleanly. The TensorFlow/automatic code-generation part is less novel but still useful, especially for quickly prototyping correlators on GPUs.

Now the soft spots, in proportion. The headline claim of "at least an order of magnitude improvement" is not supported by Table II for the lightest nuclei: 2H shows 4.35x and 3He shows 7.30x. The later CUDA/TensorFlow tables show larger gains, but those compare against serial C baselines, which is not a fair state-of-the-art comparison. The bigger worry is the truncation claim. The extra order-of-magnitude speedup from keeping only the top 0.05%–20% of terms rests entirely on the empirical stability of the dominant-term set across quark masses, volumes, and operators. The paper itself says "systematic uncertainties associated with dropping any terms need to be fully understood," and the fractions in Table VI are based on visual agreement of effective masses, not a quantitative error budget. Figures 7 and 8 show the distribution broadens as m_pi decreases, so transferability is not established. If the dominant index set shifts with mass or volume, the precomputed truncation, the extra speedup, and the spin-color symmetry conjecture all collapse. The main minimal-set algorithm survives, but that is a smaller claim than the one the abstract makes.

Also, no code or data is shipped, which makes independent reproduction harder. The baseline comparisons could be expanded to include recent contraction optimizers like Humphrey et al. (though the paper cites them). None of this is fatal, but it means the paper's most exciting claim is not yet supported.

Who is this for? Lattice QCD practitioners computing nuclear correlation functions. The randomized preprocessing is worth a serious referee, and the paper deserves peer review. A referee should push for controlled truncation tests (fix the index set on one ensemble, validate on independent masses/volumes), a quantitative error budget, and code release. If those are added or the claim is scaled back, this becomes a solid methods paper. As is, it is a promising but overreaching draft.

Referee Report

2 major / 5 minor

Summary. The paper reports two methods for evaluating nuclear two-point correlation functions in lattice QCD. The first is an optimized determinant algorithm: the correlator is written as a sum of products of per-flavor determinants [Eq. (15)], and a randomized preprocessing step based on Schwartz-Zippel identity testing identifies, once per interpolating operator, a 'minimal set' of determinant index tuples that is claimed to be independent of the propagator and of the lattice volume. The second method uses TensorFlow/opt_einsum to generate and automatically optimize tensor contractions of the block algorithm and to execute them on GPUs. The authors apply the methods to 2H, 3He, 4He, and 7Li point-sink and plane-wave-sink correlators at several lattice volumes, report wall-clock timings for serial-C, OpenMP, CUDA, and TensorFlow implementations, and extract effective masses and finite-volume energy shifts at two unphysical quark masses. They also observe that the terms contributing to the sums follow an extreme distribution, such that a small top fraction (0.05%-20%) reportedly reproduces the full effective masses, and conjecture a spin-color symmetry.

Significance. The core algorithmic contribution is sound and likely useful: the flavor factorization leading to O(nu^3+nd^3+ns^3) determinant evaluation is clean, and the randomized preprocessing has a rigorous probabilistic error bound (Appendix C) and is genuinely a one-time cost for a given operator. The GPU/TensorFlow implementation is a practical engineering contribution that will be of interest to the lattice community. The observed dominance of a small fraction of terms is intriguing and, if quantitatively controlled, could enable substantial additional savings. However, the paper's headline 'order of magnitude' claim is overstated for the light nuclei, and the truncation-based speedup and symmetry conjecture are not yet backed by a quantitative systematic analysis. Overall, the work is a solid incremental algorithmic advance with real but uneven speedups, not yet a demonstrated order-of-magnitude improvement across the entire range claimed.

major comments (2)

[Abstract; Sec. VI A; Tables II-IV; Fig. 1] The headline claim of 'at least an order of magnitude improvement over existing algorithms' conflates algorithmic improvement with hardware acceleration and is not met for the lighter nuclei. In the algorithmic comparison of Fig. 1 (same point-sink setup), the determinant algorithm is only about two times faster than the block algorithm for 2H and 3He; Table II shows the optimized-determinant speedup over the vanilla determinant is 4.35x for 2H and 7.30x for 3He. The large ratios in Tables III and IV, such as SERIAL C versus CUDA (~200x), are dominated by GPU-versus-CPU differences and do not quantify an algorithmic improvement. The stated level of speedup is therefore valid only for 4He (27.89x) and 7Li (24.72x); the claims should be qualified accordingly.
[Sec. VI C; Table VI; Appendix D] The claim that keeping only the top 0.05%-20% of terms reproduces the full correlator is based on visual agreement of effective masses in plateau regions, with no quantitative error budget; the manuscript itself states in Sec. VI C that 'systematic uncertainties associated with dropping any terms need to be fully understood.' Moreover, the required fraction varies with quark mass and nucleus (Table VI; Figs. 7-8), and no test establishes that a top-term index set identified on one small lattice or ensemble remains sufficient on an independent ensemble at another mass, volume, or operator. Because the additional 'order of magnitude' speedup and the spin-color symmetry conjecture in Sec. VI C depend on this truncation, a quantitative validation or an explicit demarcation of the claim as a preliminary observation is needed.

minor comments (5)

[Sec. VI C, Fig. 7 caption] The caption says 'Distribution plots for 2He' but the text and context refer to the deuteron, so this should read '2H'.
[Sec. IV] The sentence 'We provide the snippet of the one nucleon code in the Appendix C' appears to be a wrong cross-reference; the TensorFlow code snippet is shown in the main text and the code-generation details are in Appendix B, while Appendix C is about identity testing.
[Table II] The entry '2 × 1026a' is ambiguous in print; the footnote says '1026 terms', so please format the table to avoid the reader interpreting '1026' as 10^26.
[Fig. 1] Because the block-algorithm comparison is a central quantitative claim, please add the numerical timing values or label the bars explicitly; the log-scale plot alone does not allow the reader to verify the stated factors of ~2 and ~10^3.
[Sec. III B] The claim that the minimal set is independent of the propagator and lattice volume is argued for generic point operators; for smeared or extended operators, which are needed for the A~12 outlook, the χ indices include spatial positions, so the claim that the precomputation can be performed on 'one set of space-time points' needs a precise definition of the smearing region and a justification that the resulting set is independent of lattice size.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the determinant/minimal-set derivation is self-contained, and the top-term truncation caveat is an acknowledged limitation rather than a circular reduction.

full rationale

The paper's core derivation chain is self-contained. The determinant algorithm begins from Wick's theorem (Eq. 11), factorizes the flavor permutation sums (Eq. 12), and expresses the two-point function as sums of products of determinants (Eq. 15); the claimed time complexity N N' × O(n_u^3 + n_d^3 + n_s^3) is a direct consequence of evaluating the small flavor determinants separately, not an input of the calculation. The minimal determinant set is obtained by randomized identity testing (Appendix C): determinants are viewed as polynomials in the propagator entries, and equality on random integer instantiations implies polynomial identity with the bounded Schwartz-Zippel error probability, so the resulting index set is genuinely propagator-independent. This step is backed by an external, parameter-free mathematical theorem, not by a self-citation. The self-citations in the paper (e.g., Refs. [56,59,60]) are background lattice-QCD results and are not load-bearing for the algorithmic claims. The only potentially questionable step is the 'top few percent terms' truncation (Sec. VI C, Table VI, Appendix D), where required fractions are selected after comparing truncated and full effective masses on the same ensembles and then proposed as a cheap precomputable set. However, the paper explicitly discloses that this is not yet a controlled systematic statement: 'The improvement observed is, so far, numerical and systematic uncertainties associated with dropping any terms need to be fully understood.' That is an acknowledged limitation and a transferability risk, not a constructively circular reduction of the kind where an output equals an input by definition. No equation is used as both premise and conclusion, and no fitted parameter is renamed as a prediction. Accordingly, no circularity step is identified and the score is 0.

Assumptions & free parameters 1 free parameters · 6 assumptions · 1 invented entities

The central algorithm rests on standard linear algebra, Wick's theorem, and a randomized identity-testing lemma, all of which are standard. The main domain assumption is the propagator-independence of the minimal determinant set. The most fragile ingredient is the empirical truncation-fraction assumption, which is post hoc and explicitly not yet systematically validated. No fitted physical constants enter the algorithmic derivation, but the per-case term-fraction thresholds are free parameters chosen to make truncation work.

free parameters (1)

per-case term-fraction threshold = 0.05%, 0.3%, 1%, 5%, 10%, 20% depending on nucleus, quark mass, and sink type
Table VI assigns each nucleus/quark-mass/sink case a required fraction chosen by comparing effective masses from truncated versus full sums. The fractions are tuned post hoc to reproduce the full correlator and are not predicted by the algorithm.

assumptions (6)

standard math Wick's theorem and Grassmann anti-commutation of quark fields
Used in Eq. (11) to expand the path-integral over quark fields into permutation sums over propagators.
standard math The flavor-separated permutation sums factorize into determinants
Equations (12)-(15) express the sums over up, down, and strange permutations as products of determinants of quark propagator matrices.
standard math Schwartz-Zippel identity testing over random integer matrices detects determinant polynomial equality
Appendix C uses this lemma to identify redundant determinant computations and provides the error bound used for the minimal set.
domain assumption The minimal set of determinant index tuples is independent of lattice volume, gauge configuration, and time slice
Sec. III B asserts this propagator independence; it follows from polynomial identity but is a structural assumption about the interpolating operators and their symmetry.
ad hoc to paper Truncating the sum to largest absolute terms is a controlled approximation
Sec. VI C uses this assumption to claim that a small fraction of terms reproduces full correlators, while explicitly deferring systematic uncertainty study to future work.
domain assumption MILC HISQ ensembles with m_u=m_d=m_s or m_c and overlap valence propagators are a valid testbed for algorithmic benchmarking
Sec. VI B uses these ensembles and propagators for the numerical demonstrations and timing measurements.

invented entities (1)

As-yet-unidentified spin-color symmetry
purpose: Proposed to explain why only a small fraction of spin-color combinations dominates nuclear correlation functions
The paper speculates that an unidentified symmetry forces specific spin-color combinations to dominate, but provides no falisifiable handle outside the same correlator data it analyzes. It states only that the observation 'hints at a potential connection to an as-yet-unidentified symmetry.'

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Cite this review

Pith. "Pith review of Nuclear correlation functions using first-principle calculations of lattice quantum chromodynamics." pith.science (2026). https://pith.science/paper/FA5ZDSOI

@misc{pith2026241108962,
  author       = {Pith},
  title        = {Pith review of: Nuclear correlation functions using first-principle calculations of lattice quantum chromodynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FA5ZDSOI}},
  note         = {Machine review of arXiv:2411.08962}
}

read the original abstract

Exploring nuclear physics through the fundamental constituents of the strong force -- quarks and gluons -- is a formidable challenge. While numerical calculations using lattice quantum chromodynamics offer the most promising approach for this pursuit, practical implementation is arduous, especially due to the uncontrollable growth of quark-combinatorics, the so-called Wick-contraction problem of nuclei. We present here two novel methods providing a state-of-the-art solution to this problem. In the first, we exploit randomized algorithms inspired from computational number theory to detect and eliminate redundancies that arise in Wick contraction computations. Our second method explores facilities for automation of tensor computations -- in terms of efficient utilization of specialized hardware, algorithmic optimizations, as well as ease of programming and the potential for automatic code generation -- that are offered by new programming models inspired by applications in machine learning (e.g., TensorFlow). We demonstrate the efficacy of our methods by computing two-point correlation functions for Deuteron, Helium-3, Helium-4 and Lithium-7, achieving at least an order of magnitude improvement over existing algorithms with efficient implementation on GPU-accelerators. Additionally, we discover an intriguing characteristic shared by all the nuclei we study: specific spin-color combinations dominate the correlation functions, hinting at a potential connection to an as-yet-unidentified symmetry in nuclei. Moreover finding them beforehand can reduce the computing time further and substantially. Our results, with the efficiency that we achieved, suggest the possibility of extending the applicability of our methods for calculating properties of light nuclei, potentially up to A ~12 and beyond.

Figures

Figures reproduced from arXiv: 2411.08962 by the authors.

**Figure 2.** Schematic sketches showing the representative Wick-contraction diagrams for the two [PITH_FULL_IMAGE:figures/full_fig_p030_2.png] view at source ↗

**Figure 3.** The two-point correlation functions of 1H, 2H, 3He, 4He and 7Li on a 483 × 144 lattice ensemble (wall source, lattice spacing a ∼ 0.0582 fm with a volume ∼ (2.8 fm)3 ×8.4 fm). The left pane is for the case with the quark masses set at the physical strange quark mass (mu = md = ms; pseudoscalar meson mass, mps = 688 MeV, Nconfig = 168) while the right pane corresponds to that at the physical charm quark mass (mu = md… view at source ↗

Figures from the paper (20 more)

**Figure 4.** Figure 4: Effective mass plots for 1H, 2H, 3He, 4He and 7Li (bottom to top) at the strange and charm quark masses (left: mu = md = ms and right: mu = md = mc), corresponding to the correlators shown in [PITH_FULL_IMAGE:figures/full_fig_p033_4.png]

**Figure 5.** Figure 5: Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p034_5.png]

**Figure 6.** Figure 6: Left: The distributions of non-zero of terms that enters in Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p036_6.png]

**Figure 7.** Figure 7: Distribution plots for 2He, similar as [PITH_FULL_IMAGE:figures/full_fig_p038_7.png]

**Figure 8.** Figure 8: Left: Histogram plots corresponding to Fig. [PITH_FULL_IMAGE:figures/full_fig_p038_8.png]

**Figure 9.** Figure 9: Effective mass of 4He obtained using all and only top 0.3%, and 0.05% terms (absolute large) of the correlation functions. This result is obtained using a 403 × 64 ensemble for the case of mu = md = mc. Right: Same results for the case of mu = md = ms. Operator set up:…

**Figure 10.** Figure 10: Schematic diagram showing CUDA-C implementation of the determinant algorithm for [PITH_FULL_IMAGE:figures/full_fig_p044_10.png]

**Figure 11.** Figure 11: Automatic code generation (using the opt einsum package along with TensorFlow) for a single proton. 1 \ documentclass [ a 4 papper ,11 pt ]{ article } 2 \ usepackage { amsmath } 3 \ RequirePackage [ hyperindex ]{ hyperref } 4 \ begin { document } 5 \ begin { multline …

**Figure 12.** Figure 12: LATEX code for the expression in eq. (B1) Similarly, for other nuclei we can print the optimized codes automatically. However, those [PITH_FULL_IMAGE:figures/full_fig_p048_12.png]

**Figure 13.** Figure 13: The distributions of non-zero of terms that enters in Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p052_13.png]

**Figure 14.** Figure 14: The distributions of non-zero of terms that enters in Eqs. ( [PITH_FULL_IMAGE:figures/full_fig_p053_14.png]

**Figure 15.** Figure 15: Comparison of distribution data of 2H and 4He at three different quark masses. Distribution is more extreme for 4He than that of 2H and that pattern also increases as the quark mass increases. correlation function is calculated by contracting the block tensors and su…

**Figure 16.** Figure 16: Effective mass of 2H obtained using all and top certain fraction of terms (absolute large) of the correlation functions. Left: For the case of mu = md = mc on a 403 × 64 lattice ensemble with Nconfig = 100. Data for 20% terms are shifted slightly to the right to avoid…

**Figure 17.** Figure 17: Left: same as Fig [PITH_FULL_IMAGE:figures/full_fig_p055_17.png]

**Figure 18.** Figure 18: Histogram plot (bin size 0.01) showing the distribution of terms (normalized with [PITH_FULL_IMAGE:figures/full_fig_p055_18.png]

**Figure 19.** Figure 19: Effective masses of 3He (left side) and 4He (right side), obtained using all and top several other percentage of terms (absolute large) of the correlation functions. Results for 3He (left) is for the case of mu = md = mc, on a 403 × 64 ensemble with Nconf = 100, while…

**Figure 20.** Figure 20: Effective masses of 7Li obtained using all and top several other percentage of terms (absolute large) of the correlation functions. The left pane corresponds to the case of mu = md = ms, while the right pane is for the case of mu = md = mc. Lattice and operator set up…

**Figure 21.** Figure 21: Three-dimensional plots showing the distribution of terms constructing two-point corre [PITH_FULL_IMAGE:figures/full_fig_p058_21.png]

**Figure 22.** Figure 22: Colormap plots corresponding to Fig [PITH_FULL_IMAGE:figures/full_fig_p058_22.png]

**Figure 23.** Figure 23: Colormap plots showing the distribution of top 1% terms constructing two-point cor [PITH_FULL_IMAGE:figures/full_fig_p059_23.png]

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Forward citations

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(B1) Here the subscript “0” refers to the indices of the source point. 1 + \ 2 oe . contract ( eps , [ ’ i_0 ’ , ’ j_0 ’ , ’ k_0 ’] , C , [ ’ beta_0 ’ , ’ gamma_0 ’] , eps , [ ’ i_ 0_ pr i me ’ , ’ j _0 _p ri me ’ , ’ k _0 _p ri me ’] , C , [ ’ b e t a _ 0 _ p r i m e ’ , ’ g a m m a _ 0 _ p r i m e ’] , S , [ ’ x_0 ’ , ’ i_0 ’ , ’ alpha_0 ’ , ’ i _0 _p r...
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Reviewed August 12, 2026 · model on record in the stance chip above.

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1 + \ 2 oe

(B1) Here the subscript “0” refers to the indices of the source point. 1 + \ 2 oe . contract ( eps , [ ’ i_0 ’ , ’ j_0 ’ , ’ k_0 ’] , C , [ ’ beta_0 ’ , ’ gamma_0 ’] , eps , [ ’ i_ 0_ pr i me ’ , ’ j _0 _p ri me ’ , ’ k _0 _p ri me ’] , C , [ ’ b e t a _ 0 _ p r i m e ’ , ’ g a m m a _ 0 _ p r i m e ’] , S , [ ’ x_0 ’ , ’ i_0 ’ , ’ alpha_0 ’ , ’ i _0 _p r...

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Thus, for greater accuracy and simplicity, it is better to take the set C from which evaluation points are drawn to be a set of integers, say from 1 to |C|

Note that the coefficients of the determinants above are integers (even though the matrix entries in S may be arbitrary complex numbers). Thus, for greater accuracy and simplicity, it is better to take the set C from which evaluation points are drawn to be a set of integers, say from 1 to |C|. This ensures that all the arithmetic performed in the tests is...

[3] [3]

In particular, if one uses machine integers, one may suspect that some of the test results above may be spurious

However, since we would typically like |C| to be about ten times nf , a potential problem with large nf (roughly when nf ≥ 7) may be the possibility of integer overflow (if machine integer types are used) or slow speed and large memory use (if exact rational arithmetic is used). In particular, if one uses machine integers, one may suspect that some of the...

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