scholarly journals On Zeros of a Polynomial in a Finite Grid

2018 ◽  
Vol 27 (3) ◽  
pp. 310-333 ◽  
Author(s):  
ANURAG BISHNOI ◽  
PETE L. CLARK ◽  
ADITYA POTUKUCHI ◽  
JOHN R. SCHMITT

A 1993 result of Alon and Füredi gives a sharp upper bound on the number of zeros of a multivariate polynomial over an integral domain in a finite grid, in terms of the degree of the polynomial. This result was recently generalized to polynomials over an arbitrary commutative ring, assuming a certain ‘Condition (D)’ on the grid which holds vacuously when the ring is a domain. In the first half of this paper we give a further generalized Alon–Füredi theorem which provides a sharp upper bound when the degrees of the polynomial in each variable are also taken into account. This yields in particular a new proof of Alon–Füredi. We then discuss the relationship between Alon–Füredi and results of DeMillo–Lipton, Schwartz and Zippel. A direct coding theoretic interpretation of Alon–Füredi theorem and its generalization in terms of Reed–Muller-type affine variety codes is shown, which gives us the minimum Hamming distance of these codes. Then we apply the Alon–Füredi theorem to quickly recover – and sometimes strengthen – old and new results in finite geometry, including the Jamison–Brouwer–Schrijver bound on affine blocking sets. We end with a discussion of multiplicity enhancements.

2021 ◽  
Author(s):  
Jinyu Zuo ◽  
Natalia Schmid

Daugman’s design of IrisCode continues fascinating the research world with its practicality, efficiency, and outstanding performance. The limits of Daugman’s recognition system, however, remain unquantified. Multiple approaches to scale performance have been explored in the past. Despite them, the problem of finding the capacity of IrisCode remains open.<br>In an attempt to fill the gap in understanding the performance limits of Daugman’s algorithm, we turn to an analysis of the relationship between the size of the population that the IrisCode can effectively cover and the iris sample quality. Given Daugman’s IrisCode algorithm, the problem of finding its capacity is cast as a basic Rate-Distortion/Channel Coding problem. The Hamming, Plotkin, and Elias-Bassalygo upper bounds on the population of a binary code under the constraint of a minimum Hamming Distance between any two codewords is applied to relate the number of iris classes that the IrisCode algorithm can sustain and the quality of iris data expressed in terms of Hamming Distance.<br><br>


2021 ◽  
Author(s):  
Jinyu Zuo ◽  
Natalia Schmid

Daugman’s design of IrisCode continues fascinating the research world with its practicality, efficiency, and outstanding performance. The limits of Daugman’s recognition system, however, remain unquantified. Multiple approaches to scale performance have been explored in the past. Despite them, the problem of finding the capacity of IrisCode remains open.<br>In an attempt to fill the gap in understanding the performance limits of Daugman’s algorithm, we turn to an analysis of the relationship between the size of the population that the IrisCode can effectively cover and the iris sample quality. Given Daugman’s IrisCode algorithm, the problem of finding its capacity is cast as a basic Rate-Distortion/Channel Coding problem. The Hamming, Plotkin, and Elias-Bassalygo upper bounds on the population of a binary code under the constraint of a minimum Hamming Distance between any two codewords is applied to relate the number of iris classes that the IrisCode algorithm can sustain and the quality of iris data expressed in terms of Hamming Distance.<br><br>


2013 ◽  
Vol 411-414 ◽  
pp. 1994-1997
Author(s):  
Yan Li ◽  
Wen Ju Zhao ◽  
Zhen Hua Zhou

This paper defined the full connect map and contact surface, and proposed a new map complexity measure, and compared with measurement methods based on Hamming distance and relative Hamming distance. We further research on the relationship between the complexity measure and the map connectivity. The complexity measures based on Hamming distance and contact surface are applicable to full connectivity map, and the new measurement can reflects the difficulty of the pathfinding algorithm more accurately, especially in a higher complexity.


2018 ◽  
Vol 28 (2) ◽  
pp. 253-279
Author(s):  
O. GEIL ◽  
U. MARTÍNEZ-PEÑAS

We upper-bound the number of common zeros over a finite grid of multivariate polynomials and an arbitrary finite collection of their consecutive Hasse derivatives (in a coordinate-wise sense). To that end, we make use of the tool from Gröbner basis theory known as footprint. Then we establish and prove extensions in this context of a family of well-known results in algebra and combinatorics. These include Alon's combinatorial Nullstellensatz [1], existence and uniqueness of Hermite interpolating polynomials over a grid, estimations of the parameters of evaluation codes with consecutive derivatives [20], and bounds on the number of zeros of a polynomial by DeMillo and Lipton [8], Schwartz [25], Zippel [26, 27] and Alon and Füredi [2]. As an alternative, we also extend the Schwartz-Zippel bound to weighted multiplicities and discuss its connection to our extension of the footprint bound.


Blood ◽  
2019 ◽  
Vol 134 (Supplement_1) ◽  
pp. 5768-5768
Author(s):  
Adekemi Taylor ◽  
Martine Allard ◽  
Cecile Kresja ◽  
Dana Lee ◽  
Greg Slatter

Introduction: KRT-232 is a potent and selective, targeted small molecule inhibitor of human mouse double minute 2 (MDM2) homolog interactions with tumor protein 53 (p53). MDM2 prevents p53 activation and reduces p53-mediated transcription and cell cycle control. KRT-232 is under development by Kartos Therapeutics for treatment of myelofibrosis, polycythemia vera, acute myeloid leukemia (AML) and Merkel cell carcinoma (see NCT03662126, NCT03669965, NCT03787602). The KRT-232 no effect-level for in vitro inhibition of hERG function (10 μM) was approximately 147- and 73-fold greater than KRT-232 unbound Cmax concentrations for steady state doses of 240 mg and 480 mg, respectively, based on population pharmacokinetic (PK)-derived parameters for subjects with AML (Ma et al. submitted, ASH 2019). The primary objective of this analysis was to evaluate the relationship between KRT-232 plasma concentration and changes in heart rate-corrected QT interval duration (QTc) in oncology patients treated in Amgen studies 20120106 (Gluck et al. Invest New Drugs; in press, NCT01723020) and 20120234 (Erba et al. Blood Adv 2019; NCT02016729). Methods Study 20120106 was a 2-part Phase 1 dose-exploration and dose-expansion monotherapy study in advanced solid tumors or multiple myeloma. KRT-232 doses of 15 mg (n=3), 30 mg (n=3), 60 mg (n=4), 120 mg (n=7), 240 mg (n=76), 300 mg (n=4), 360 mg (n=4) and 480 mg (n=6) were administered daily (QD) for 7 days in 21-day cycles. Subjects received up to 31 cycles of treatment. Study 20120234 was a Phase 1b study evaluating KRT-232 alone and in combination with trametinib in relapsed/refractory AML. Subjects received the following KRT-232 doses: 60 mg (n=14; n=10 co-administered with 2 mg trametinib daily [excluded from C-QTc analysis]); n=4 as single agent), 90 mg (n=4), 180 mg (n=5), 240 mg (n=3), and 360 mg (n=10). Doses were administered QD for 7 days in 14-day cycles. Subjects received up to 46 cycles of treatment. In both studies, time-matched PK and ECG measurements were collected intensively during Cycle 1 and less frequently at other visits. Triplicate 12-lead ECG data (N=3) were read by a central laboratory. A linear mixed effects model using R (v 3.5.2) was used to analyze the relationship between KRT-232 plasma concentrations and the QT interval corrected using Fridericia's method (QTcF). Effects of baseline QTcF, study, sex and tumor type on C-QTc were investigated. The upper bound of 2-sided 90% CIs for the mean QTcF change from baseline (ΔQTcF) predicted at Cmax was compared to the 10 ms threshold of regulatory concern (FDA Guidance: E14(R3) 2017; Garnett et al. Pharmacokinet Pharmacodyn 2018). Results ECG and PK data for this analysis were available from 130 subjects. The final model was a linear mixed-effect model with parameters for intercept, KRT-232 concentration-ΔQTcF slope, and baseline QTcF effect on the intercept. Diagnostic plots indicated an adequate model fit. The final C-QTc model was used to predict mean ΔQTcF and associated 2-sided 90% CI mean steady-state KRT-232 Cmax at doses up to the maximum clinical dose of 480 mg QD, in subjects with AML or solid tumors. The mean and upper bound of the 90% CI of ΔQTcF were predicted not to exceed 10 ms at doses of up to 480 mg QD in subjects with AML, multiple myeloma or solid tumors. Mean (90% CI) predicted ΔQTcF values at 480 mg QD were 2.040 (0.486, 3.595) ms for subjects with solid tumors and 4.521 (2.348, 6.693) ms for subjects with AML (Figure A). The KRT-232 concentrations at which the upper bounds of 90% CI of mean ΔQTcF are predicted to reach 10 ms and 20 ms are 4298 ng/mL and 7821 ng/mL, respectively. These concentrations are 2.2- and 4-fold higher, respectively, than the predicted mean steady-state Cmax for 480-mg KRT-232 in subjects with solid tumors, and 1.4- and 2.5-fold higher, respectively, than the corresponding mean steady-state Cmax in subjects with AML. Conclusion Since the mean and upper bound of the 90% CI of mean ΔQTcF were predicted not to exceed 10 ms at KRT-232 doses of up to 480 mg QD in solid tumor or AML patients, KRT-232 should not result in clinically meaningful QT prolongation at the doses currently under investigation in Kartos clinical trials. Disclosures Taylor: Certara Strategic Consulting: Consultancy, Employment. Allard:Certara Strategic Consulting: Consultancy, Employment. Kresja:Kartos Therapeutics: Employment, Equity Ownership. Lee:Kartos Therapeutics: Employment, Equity Ownership. Slatter:Kartos Therapeutics: Employment, Equity Ownership. OffLabel Disclosure: KRT-232 (formerly AMG 232) is a small molecule MDM2 inhibitor


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