# Richard J. Lipton's research while affiliated with Telcordia Technologies and other places

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## Publications (365)

It is shown that the Jacobian Conjecture (in all dimensions) is equivalent to the following statement: for almost all prime numbers p and each Keller map (i.e. ), the induced map is not the zero map.

Efficient secure password protocols are constructed that remain secure against offline dictionary attacks even when a large, but bounded, part of the storage of a server responsible for password verification is retrieved by an adversary through a remote or local connection. A registration algorithm and a verification algorithm accomplish the goal o...

A method and software product for limit privacy loss due to data shared in a social network, where the basic underlying assumptions are that users are interested in sharing data and cannot be assumed to constantly follow appropriate privacy policies. Social networks deploy an additional layer of server-assisted access control which, even under no a...

People, problems, and proofs are the lifeblood of theoretical computer science. Behind the computing devices and applications that have transformed our lives are clever algorithms, and for every worthwhile algorithm there is a problem that it solves and a proof that it works. Before this proof there was an open problem: can one create an efficient...

The main complexity lower and upper bound frontiers seem to be moved only once a decade lately, and Ryan Williams kicked off this decade by proving a new lower bound against Boolean circuits with modular counting. This chapter surveys his proof strategy and why he was able to break through a twenty-year barrier.

This chapter reflects on three big results from 2010, on unique games, Boolean circuits with modular counting, and lower bounds against the simplex method of linear programming. The last connects to foundational work by Klee with George Minty.

This chapter describes a breakthrough idea and result by Balcan with Avrim Blum and Anupam Gupta. Where a worst-case model is “too cold” and a random-case model is “too hot”, they formulate a natural in-between situation that arises importantly in practice. Not only do they prove new performance guarantees, they establish new ways of designing algo...

A pseudorandom generator famously designed by Noam Nisan underlies many results and problems about low space-bounded complexity classes. Matei David and two co-authors recently proved a promising extension of the generator, enabling one to analyze multi-pass computations in these classes.

This chapter discusses aspects of Chattopadhyay’s results with Avi Wigderson on representing Boolean functions by polynomials in the integers modulo a composite number. When the modulus is prime many answers are known, but the composite case brings issues at the frontier of complexity lower bounds.

In this chapter our drumbeat is that sometimes strange numbers arise in mathematics, in theory, and in even in physics. Richard Feynman was fond of some strange numbers, and we are too. There is a new polynomial time algorithm whose exponent is very large—another strange result? We will discuss the role of such strange numbers and strange results.

This chapter describes a result by Smith with Venkatesan Guruswami on security over limited channels. Possible connections are drawn again for the satisfiability problem.

What methods do trained mathematicians use to attack hard problems? What is in their bag of tricks that an amateur may not know? This chapter attempts to give general advice and categorize methods, including the possibility of broadening methods used by subfields.

This chapter presents a theorem from 1990 by Bruno Courcelle that furthers relations between logic and complexity classes that were originated by Ron Fagin. Monadic second-order logic and tree decompositions of graphs are at front and center, but lurking behind Big-Oh notations are huge constants that render the associated algorithms “galactic” unt...

Dick Karp may be said to have founded the polynomial hierarchy by showing the sweep of the polynomial-time reductions by which it is organized into levels. The chapter mainly discusses a theorem by Ravi Kannan on circuit lower bounds whose proof uses the second through fourth levels in a non-constructive manner, and asks whether it can be done more...

This chapter discusses two unexpected results by Strassen: his sub-cubic time for matrix product and his super-linear lower bound with Walter Baur on arithmetical circuit size. Possible connections are drawn to new results by Ryan and Virginia Williams involving graphs and matrices—this was a year before the latter’s improvement on Strassen’s matri...

The late Shimon Even was one of the inventors of “promise problems,” which demand correct solutions only for instances having a separate promised property. The chapter described a simple problem about exponentially long pairs of paths in graphs promised to have nothing else that is surprisingly hard to classify.

This chapter describes the theory of equations over groups, and introduces connections between algebriac groups and complexity classes uncovered by Thérien and his co-workers. The ability to represent solutions to certain equations makes a group “universal” for these classes, and we ponder exactly what obstructs solvable groups from being universal...

The famous “ken” of UNIX and C and UTF, whom Ken Regan knew beginning in 1971 from the Westfield Chess Club in New Jersey, pioneered computer chess through his program Belle and the compilation of tables giving perfect play in certain endgames. This chapter surveys data structures and algorithms for these tables.

This chapter discusses attempts over the years to get lower bounds on the NP-complete satisfiability problem. Hennie’s lower bounds for a one-way machine model helped spur still-lively research on time-space tradeoffs for satisfiability.

This chapter surveys recent results on Subhash Khot’s Unique Games Conjecture. This conjecture has been a unifying force in complexity theory, but these results hint at its being false. Special cases are suggested whose analysis may also delineate the conjecture’s practical implications.

This chapter discusses a question related to Leonardo da Vinci as an artist: can we view great proofs as great art? Four famous proofs in mathematics are compared for their context, ideas, core innovation, plan, and craft.

Ramanujan was no “amateur”, but he shared with many the lack of formal training in mathematics when he began his research. The chapter discusses traits he shared with amateurs, including difficulty writing proofs, and gives famous historical contributions by amateurs.

Hopcroft is hailed not only for his seminal work on algorithms and the foundations of complexity theory, but also for new ways of getting subfields to talk with each other. The chapter tells the story of Lipton’s involvement with Hopcroft in creating and supporting national federated conferences.

In this chapter we discuss the role of proof in mathematics and theory. A proof must convince not only the author but others; if it does not, it is not a proof. One wonderful example of this is Roger Apéry’s proof that the zeta function evaluated at 3 is irrational. This was a result that the great Leonhard Euler missed, and many at first doubted A...

“Galactic” algorithms, a term coined by Regan, are algorithms with good-looking asymptotic running times but concrete costs so high as to prevent their practical use on scales smaller than the universe. Several algorithms are critiqued from this point of view.

This chapter describes Chazelle’s concept of “Natural Algorithms” amid other uses of “natural” in mathematics and computer science. A theorem of his about equilibria in Nature exposes some of the ideas.

Dantzig was the progenitor of the simplex method for linear programming. A viewpoint in his spirit approaching P versus NP entirely via linear equations is presented and defended.

Rivest is known as the ‘R’ in the RSA cryptosystem, which after four decades has yet to find peers in simplicity, utility, and beauty. This chapter discusses similar presents in mathematics including another from Rivest about elusive Boolean functions.

This chapter continues the discussions of the three previous ones, and speculates on which other open mathematical problem should be anointed as the replacement for the Clay problem whose solution was completed by Perelman. Seven candidates are discussed.

This chapter outlines a survey paper by van den Essen on the Jacobian Conjecture, which is on Stephen Smale’s short list of important problems besides the Millennium Prize ones. A new link discovered by Wenhua Zhao and broadened by van den Essen supplies new ideas and approaches.

This chapter discusses how good we are at guessing the truth of mathematical conjectures, noting that one by Hirsch was recently disproved. Those who guess wrong have a lot of excellent company.

This chapter describes a new algorithm for the Knapsack problem that was presented at the 2010 EUROCRYPT conference. In like manner to Björklund’s paper, randomization is used to bring down the exponent of an exponential-time but possibly practical algorithm.

Sometimes the inventor of a “method” deserves more credit than the one who applies it to obtain results, but it is often harder to trace the origin of a method than a theorem. The “Coefficient Extraction Method” used by Lokshtanov and his co-authors is traced to earlier methods involving polynomial equations.

The starting quarterback of the NFL Cleveland Browns was full-time mathematics faculty at Case Western, even earning promotion to Associate Professor in 1971. Lipton took a seminar with him and tells stories of his “team teaching” methods.

This chapter discusses a heuristic technique called “linearization” that has been quite powerful in attacking cryptosystems. An attack developed by Courtois and co-workers is turned into a method for proofs in the complexity of learning Boolean functions.

This chapter discusses what it would mean to have a proof of P=NP or the opposite, apart from the major and minor claims to prove them equal or unequal in 2010. A proof should embrace its explanatory and educational value from the start, not merely be the verification of a claim.

A little-known theorem with a great name has surprisingly deep importance in group theory. This chapter presents both its proof and its potentially extandable application.

Smullyan is one of the great popular expositors of logic, as well as author of several books on puzzles and games. The probability version of Cantor’s proof is re-cast following Smullyan’s angle as a two-player game, further challenging readers who doubt the correctness of Cantor’s argument.

The name in German means “Theorem on Locations of Zeroes”, and Hilbert’s theorem remains the gold standard in determining when equations are solvable over the complex numbers. The chapter also gives an example of how effective versions of the Nullstellensatz help in finding equations to begin with.

The claim in August 2010 by Dr. Vinay Deolalikar of HP Labs to have proved P not equal to NP, which would solve one of the million-dollar Clay Millennium Prize Problems, commanded the attention of much of the world mathematical community. The popular press also picked up the larger story of people coming together to review Deolalikar’s paper during...

This chapter resumes the discussion of barriers faced by amateurs in science and technology. The famous Holywood starlet originated the idea of “frequency hopping” to solve problems of controlling devices by radio in the presence of jamming.

This chapter discusses the sensitive question of whether researchers should share their ideas, especially ideas that are not “complete.” The example of how Hamilton’s work on the Poincaré Conjecture was used by Grigori Perlman is raised before revisiting the Unique Games Conjecture.

This chapter outlines Ryan Williams’ breakthrough lower bound of the previous chapter in considerable detail. The ideas in a thirty-page paper are explained and the proof method mapped out.

This chapter outlines Björklund’s paper at the FOCS 2010 conference, which solved a decades-old problem about Hamiltonian cycles in graphs. Sums of determinants of randomly extended matrices are key to breaking a power-of-2 barrier in algorithmic running time for detecting these cycles.

This chapter describes a pretty theorem by Allender, and connects it to the previous chapters’ discussion on partial results. Lipton and Regan shared their own idea for trying to extend Allender’s theorem, but an astute reader showed why it couldn’t work.

Logical theories have their own complexity measures, for the problems of deciding whether a statement is true or finding a proof of a theorem. Tarski proved that the first-order theory of the real numbers is decidable, but left open its complexity. This chapter surveys exponential lower bounds proved subsequently for it.

This chapter discusses an intensive theorem by the combinatorial logician Martin Grohe, building on Seymour’s pathbreaking work with Neil Robertson on the complexity of properties defined by excluding graph minors. No one has ever provided a logic that captures polynomial-time decidability on graphs without referencing an ordering of the vertices;...

This chapter presents a lesser-known limitation of first-order logic. Lipton had the experience of sitting on a potential breakthrough in complexity theory if only he could encode a certain construction in first-order logic. He approached Buss, who used his background in bounded logic to recognize it as a known impossibility.

Current world championship chess matches inspired a chapter about combinatorial game theory. Berlekamp found depth in games such as “Dots and Boxes” that look easier than chess, and helped formulate central combinatorial problems via games. Games for coloring and packing problems are discussed.

Mathematics consists of definitions, equations, and proofs, and while the last two close the sale, definitions are like “Location, location, and location” in real estate. Types and pitfalls of definitions are discussed.

This chapter discusses how to claim a major result: how to write it up, and how to release it to the world. Some analogies are drawn from classic Sherlock Holmes stories and films.

Schulman was program chair of the STOC 2010 conference. This chapter presents a surprising upper bound he proved in 1999 with Sridhar Rajagopalan on the complexity of testing whether an unknown binary operator is associative.

The notion of complexity classes of problems is one of the great definition success stories in science. This chapter traces its lineage to the influence of Hartley Rogers’ famous text and papers systematizing computability theory in the 1950s and 1960s.

This chapter draws analogies between mountain climbing and solving mathematical problems. Not just the solution but also the avenue of ascent must be carefully planned, and one must also learn when to turn back and try another path.

This chapter discusses a theorem by Jech on a finitary version of the Axiom of Choice. A bootstrapping trick for constructing finite choice functions may have other applications in theory.

This chapter discusses the famous diagonal method of Georg Cantor to prove that the real numbers are uncountable. Two variants on the classic proof are presented, one involving probability.

This chapter discusses the role of intuition in mathematics: what it is, how to get it, and how to use it. After giving a famous example in number theory involving Bombieri, a proof contributed by reader Colin Reid to refute the previous chapter’s intuition about the solving power of finite groups is presented and assessed.

In this chapter we will discuss two central results of complexity theory that we know something about, thanks to the foundational work of Patrick Fischer. Yet these results have not been improved for decades, and perhaps now is the time, finally, to make some progress on them.

This chapter presents Chee Yap’s solution to a problem that had been raised on the blog in 2009. Yap improved the computation of selected digits of pi—without having to compute all previous digits—to run in logarithmic space, and used Lipton’s suggestion about measures of how far an irrational number is from being rational to prove the correctness...

This chapter ends with the theme that even when you have a proof, do you really know what you think you know? Proofs of security of quantum communication systems have proved especially liable to kinds of attacks that were underestimated or not considered in the provers’ modeling. Thus research seems unavoidably a social process, a game in which one...

A pivotal flaw in Vinay Deolalikar’s claimed proof of P not equal to NP, the subject of the opening chapter, was an incorrect assertion about closure under projections. The great Henri Lebesgue erred on the same score over a century ago. This chapter contrasts Deolalikar’s situation with a famous complexity lower bound in which closure under projec...

This chapter discusses a survey article by Koblitz whose title speaks all: “The Brave New World of Bodacious Assumptions in Cryptography.” The issue is that sometimes those who prove security of systems based on hardness assumptions overlook attacks that are practical but do not fit their model.

This chapter discusses notation in mathematics and computer science theory, and how notation can affect our thinking. Kenneth Iverson’s APL programming language is introduced as an example, and critiques are given of notation by famous mathematicians and physicists.

This chapter presents Lipton’s own joint work with Atish Das Sarma on a famous open problem of Tutte, and a new approach to the problem. Tutte famously broke a claimed proof of the Four-Color Map Theorem with an unexpected counterexample graph, but some of his own conjectures about similar graphs retain their difficulty and mystery to this day.

This chapter discusses some problems about solving equations, ones of a type that are not as well known as polynomial equations. These emerge in the theory of harmonic functions, and are better for analysing Einstein’s phenomenon of gravitational lensing than polynomial equations.

This chapter describes some connections between problems in mathematics that are surprising. Seeing connections was a great strength of Nash, and new ideas on Nash equilibria in non-zero-sum games have surprisingly come from complexity and cryptography.

Now Virginia Vassilevska-Williams as the wife of Ryan of Chapter 24, she has revived several problems in graph and matrix combinatorics with unexpected algorithms. This chapter surveys her paper at the AAAI 2010 conference on the complexity of deciding, given rankings of players in a tournament and the outcomes of some played matches, it is possibl...

We give a self-reduction for the Circuit Evaluation problem (CircEval) and prove the following consequences.
Amplifying size–depth lower bounds. If CircEval has Boolean circuits of n
k
size and n
1−δ
depth for some k and δ, then for every
${\epsilon > 0}$
, there is a δ′ > 0 such that CircEval has circuits of
${n^{1 + \epsilon}}$
size and
$...

We give a self-reduction for the Circuit Evaluation problem (CircEval), and prove the following consequences. · Amplifying Size-Depth Lower Bounds. If CIRCEVAL ϵ SIZEDEPTH [nk, n1-δ] for some k and δ, then for every ε >; 0, there is a δ >; 0, there is a δ' >; 0 such that CIRCEVAL ϵ SIZEDEPTH [nk, n1-δ']. Moreover, the resulting circuits require onl...

In this note we investigate the stability of game theoretic and economic solution concepts under input perturbations. The existence of many of these solutions is proved using fixed point theorems. We note that fixed point theorems, in general, do not provide stable solutions, and we show how some commonly studied solution concepts are indeed not st...

Consider the following problem. A seller has infinite copies of n products
represented by nodes in a graph. There are m consumers, each has a budget and
wants to buy two products. Consumers are represented by weighted edges. Given
the prices of products, each consumer will buy both products she wants, at the
given price, if she can afford to. Our o...

The standard simulation of a nondeterministic Turing machine (NTM) by a deterministic one essentially searches a large bounded-degree graph whose size is exponential in the running time of the NTM. The graph is the natural one defined by the configurations of the NTM. All methods in the literature have required time linear in the size S of this gra...

We identify a sub-class of BQP that captures certain structural commonalities
among many quantum algorithms including Shor's algorithms. This class does not
contain all of BQP (e.g. Grover's algorithm does not fall into this class). Our
main result is that any algorithm in this class that measures at most O(log n)
qubits can be simulated by classic...

We show that the set of all functions is equivalent to the set of all symmetric functions (possibly over a larger domain)
up to deterministic time complexity. In particular, for any function f, there is an equivalent symmetric function f
sym such that f can be computed from f
sym and vice-versa (modulo an extra deterministic linear time computation...

This talk will discuss some recent results on the complexity theory of quantum computing. The long term goal is to understand
exactly the relationship between the new quantum complexity classes and the classic ones. It is assumed that you are not an expert in: quantum flavordynamics, quantum geometrodynamics, quantum hydrodynamics, quantum magnetod...

We study a new model of computation called stream checking on graph problems where a space-limited verifier has to verify a proof sequentially (i.e., it reads the proof as a stream). Moreover, the proof itself is nothing but a reordering of the input data. This model has a close relationship to many models of computation in other areas such as data...

— The study of computing skylines and their variants has received considerable attention in recent years. Skylines are essentially sets of most interesting (undominated) tuples in a database. However, since the number of tuples in a skyline is often too large to be useful to potential users, much research effort has been devoted to identifying a sm...

We propose the k-representative regret minimization query (k-regret) as an operation to support multi-criteria decision making. Like top-k, the k-regret query assumes that users have some utility or scoring functions; however, it never asks the users to provide such functions. Like skyline, it filters out a set of interesting points from a potentia...

George Karakostas and Anastasios Viglas are two excellent theorists, who both have their Ph.D.’s from the Princeton computer science department. I was the Ph.D. advisor of Viglas, and later had the pleasure of working with both of them on several projects.

Ed Clarke, a Turing Award winner, was at a recent NSF workshop on design automation and theory. Ed has, for approximately three decades, worked successfully on advancing the state of the art of verification, especially that of hardware.

Neil Immerman and Robert Szelepcsényi are two computer scientists who are famous for solving a long standing open problem [69, 126]. What is interesting is that they solved the problem at the same time, in 1987, and with essentially the same method. Independently. Theory is like that sometimes.

Alan Turing is of course famous for his machine based notion of what computable means—what we now call, in his honor, Turing Machines. His model is the foundation on which all modern complexity theory rests; it is hard to imagine what theory would be like without his beautiful model. His proof that the Halting Problem is impossible for his machines...

Russell Impagliazzo is one of the most original thinkers in complexity theory, and is famous for many wonderful things. One of his papers is on five worlds [70]. The worlds are: Algorithmica, Heuristica, Pessiland, Minicrypt, and Cryptomania—see his paper for the definitions.

Bob Sedgewick is an expert on the analysis of algorithms-one of the best in the world. This is not too surprising since he is a Ph.D. student of Donald Knuth. Bob is also famous for his textbooks on algorithms, and more recently his monograph on the analysis of algorithms with Philippe Flajolet. This [48] is a required book for anyone trying to und...

Zeke Zalcstein worked, before he retired a few years ago, on the boundary between mathematics and computational complexity. He started his career in mathematics, and then Zeke moved into computational complexity. His Ph.D. advisor was John Rhodes, who is an expert on many things, including the structure of semigroups; Rhodes has recently worked on...

He use to walk the rolling grounds of the Institute for Advanced Studies with Albert, but Einstein died last spring. Today Kurt Gödel is walking alone through the falling snow, and he is thinking. Gödel is the greatest logician of his time, perhaps of all time, yet he is deeply troubled. He has found a problem he cannot solve—this does not happen o...

Gil Kalai is one of the great combinatorialists in the world, who has proved terrific results in many aspects of mathematics: from geometry, to set systems, to voting systems, to quantum computation, and on. He also writes one of the most interesting blogs in all of mathematics named Combinatorics and more; I strongly recommend it to you.

Walter Savitch did seminal work in computational complexity theory, and has made many important contributions. For all his other wonderful work, he is undoubtedly best known for his beautiful result on the relationship between deterministic space and non-deterministic space.

David Letterman is not a theorist, but is an entertainer. I do not watch his show—on too late—but I love to read a recap of his jokes in the Sunday New York Times. He is of course famous for his “Top Ten Lists.”

Michael Rabin is one of the greatest theoreticians in the world. He is a Turing Award winner, with Dana Scott in 1976, a winner of the Paris Kanellakis Award, a winner of the EMET Prize, and many others. What is so impressive about Rabin’s work is the unique combination of depth and breadth; the unique combination of solving hard open problems and...

Anne Condon is a theoretician who did fundamental work early in her career in complexity theory. She then moved into the area of DNA computing, and next to the more general area of algorithms for biology. Her work in all these areas is terrific, and she is a wonderful person to work with on any project. I had the privilege of working with her on a...

Landon Clay is the Boston businessman who created the Clay Institute and the associated Millennium Prize Problems, one of which is P=NP. The solver(s) of P=NP will get, after certain rules are fulfilled, a cool one million dollars. That is a lot of money, but somehow juxtaposed next to current bank bailout and other loans measured in the trillions...

Lance Fortnow is one of the top complexity theorists in the world, and has proved fundamental results in many aspects of complexity theory. One example, is his famous work on interactive proofs and the complexity of the permanent with László Babai, which paved the way for the later result of Adi Shamir that IP = PSPACE [12, 121]. More recently, he...

Paul Erdős needs no introduction to our community, but I will give him one anyway. He won the AMS Cole Prize in 1951 and the Wolf Prize in 1983. He is famous for so many things: an elementary proof of the Prime Number Theorem, the solver of countless open problems, the creator of countless more open problems, and his ability to see to the core of a...

Dick Karp and I first met at the IFIP Congress in Stockholm in August of 1974. I recall the meeting vividly. I was, then, a freshly minted assistant professor at Yale Computer Science Department, when I met him. Dick was quite friendly and gracious towards me and I have ever since considered him a close friend. Actually the truth is I was a “Gibbs...

David Barrington is famous for solving a long standing open conjecture. What I love about his result is that not only was it open for years, not only is his solution beautiful, but he also proved it was false. Most of us who had worked on the problem guessed the other way: we thought for sure it was true. Again conventional wisdom was wrong. This m...

Pierre-Simon Laplace, was a famous mathematician from the 18th
century, who is known for many things. How about having a transform named after you—the Laplace transform—pretty cool. He is also famous for posing the scientific question, “what is the probability the sun will rise tomorrow?”
Today I will discuss our version of his question: “What is t...

Ron Graham is part mathematician, part computer scientist, part juggler, part magician-he is one of the most talented people I know. He has done terrific work in the theory of computation, and in this chapter I will talk about just one of his many great results.

Lenore Blum is a computer scientist who is now at Carnegie-Mellon University. She is famous for her work with Steve Smale and Michael Shub on the “The P=NP Question Over The Reals” [18]. This important work shows that there is a version of P=NP for computations over the real numbers. Also she is one of the national leaders in helping get more women...

Ashok Chandra is a famous theorist who has done seminal work in the many parts of the foundations of complexity theory. His early work was in program schemas, an area that was central to theory in the 1970’s. Chandra then moved into other areas of theory, and among other great things co-invented the notion of Turing Machine Alternation—see Chp. 12.

## Citations

... Within a soccer team (HV Ribeiro et al 2010), baseball, American football league, and others, the members know that the opponents, taking advantage of their skills will do everything they can to get the triumph in a competition, hence it is transparent to the participants that everyone play against each other, in other words at the end of the race is recognized as winner to the team that generates the highest score taking the circumstances of each sport, with exceptions such as the contest of diving. The sports games have a number of complications in practice (Bregie et al 2011), starting with the selection of the best players (Lipton 2013) to participate in a tournament taking into consideration the target to win a contest. ...

... Indeed, Strassen was trying to prove, by process of (intelligently exhaustive) elimination, that such an algorithm could not exist (e.g., [Lan08, Remark 1.1.1] or [LR13]). In his paper it is presented as follows, which "one easily sees" [Str69,p. ...

Reference: Designing Strassen's algorithm

... The min-gap pair needed to implement a min-gap test can be identified in optimal expected O(N ) time and space using Rabin's randomized closestpair algorithm [22,23]. Unlike the Gonzalez algorithm for max-gap, Rabin's algorithm generalizes efficiently to higher dimensions. ...

... Hence the comparison ofX andŶ can be done in time O(N M ). It is known that if there existed faster methods for obtaining the automata intersection, then significant improvements would be implied to many long standing open problems [20]. Hence an immediate reduction to the problem of NFA intersection does not particularly help. ...

Reference: Comparing Degenerate Strings

... But, at the moment there is no a necessary and sufficient condition for which the real Jacobian conjecture holds. In addition, there are also many other partial results on both Jacobian and strong Jacobian conjectures, and on their relations with other subjects, see e.g. the survey papers by Bass et al [6] in 1982 and Essen [23] in 2000, and the recent publications [5,7,13,14,19,21,24,26,27,29,32,35,38,40,47,48,49,50,55,56]. On the history of the Jacobian and strong Jacobian conjectures we should also mention the papers by Abhyankar and Moh [3] in 1975 for two dimensional case, by Wang [51] in 1980 for the map F with degrees no more than 2, by Bass et al [6] in 1982, Yagzhev [54] in 1980 and Drużkowski [20] in 1983 via reduction of degrees of the polynomial maps, and by Hubbers [30] on cubic maps in dimension 4. ...

Reference: Jacobian conjecture in $\mathbb R^2

... Afin de trouver les coefficients des polynômes R(s) et S(s) il suffit d'inverser la matrice de Sylvester.Nous pouvons ici constater que la taille de la matrice de Sylvester dépend de l'ordre du système n et du degré relatif δ 0 entre les polynômes R(s) et S(s). L'un des algorithmes d'inversion matricielle les plus rapides et applicables aété proposé par Strassen dans[64] (il existe des algorithmes plus rapides, comme celui de Coppersmith-Winograd[65], mais qui ne sont pas facilement implantables dans la pratique, ces algorithmes sont dits galactiques[66])[64]. Sa complexité dans notre cas est deO((2n + δ 0 ) log 2 (7) ) ≈ O((2n + δ 0 ) 2.807 ). ...

... Theorem 5.8 (Corollary of Theorem 3.1 from [LW12]). There exists a (polylog-time uniform) universal circuit family {C n } of size O (n · poly log(n)), such that for any bounded fan-in circuit ...

Reference: Outcome Indistinguishability

... Lipton-Williams [LW13]. If there is ε > 0 such that for every δ > 0 we have CircEval / ∈ Size-Depth[n 1+ε , n 1−δ ], then for every k ≥ 1 and γ > 0 we have CircEval / ∈ Size-Depth[n k , n 1−γ ] (in particular P NC). ...

... The notion of perturbation stability we consider in our paper is also related to the stability notions considered by Lipton et al. [29] for economic solution concepts. The main focus of their work was understanding whether for a given solution concept or optimization problem all instances are stable. ...

Reference: Nash equilibria in perturbation-stable games

... Therefore, it is natural to consider hierarchies of LP relaxations such as the Sherali-Adams hierarchy [32] (see [11] for a general survey and [14] [33] for recent algorithmic results using the Sherali-Adams hierarchy). Especially, Chalermsook et al. [5] recently showed that there is a FPTAS when the graph has bounded treewidth, based on the Sherali-Adams hierarchy. They further asked whether the Sherali-Adams hierarchy can be used to yield an approximation ratio better than 1 4 in general case. ...