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Joseph Mavericks

Joseph Mavericks

3 years ago

5 books my CEO read to make $30M

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Looi Qin En

Looi Qin En

3 years ago

I polled 52 product managers to find out what qualities make a great Product Manager

Great technology opens up an universe of possibilities.

Need a friend? WhatsApp, Telegram, Slack, etc.

Traveling? AirBnB, Expedia, Google Flights, etc.

Money transfer? Use digital banking, e-wallet, or crypto applications

Products inspire us. How do we become great?

I asked product managers in my network:

What does it take to be a great product manager?

52 product managers from 40+ prominent IT businesses in Southeast Asia responded passionately. Many of the PMs I've worked with have built fantastic products, from unicorns (Lazada, Tokopedia, Ovo) to incumbents (Google, PayPal, Experian, WarnerMedia) to growing (etaily, Nium, Shipper).

TL;DR:

  • Soft talents are more important than hard skills. Technical expertise was hardly ever stressed by product managers, and empathy was mentioned more than ten times. Janani from Xendit expertly recorded the moment. A superb PM must comprehend that their empathy for the feelings of their users must surpass all logic and data.

  • Constant attention to the needs of the user. Many people concur that the closer a PM gets to their customer/user, the more likely it is that the conclusion will be better. There were almost 30 references to customers and users. Focusing on customers has the advantage because it is hard to overshoot, as Rajesh from Lazada puts it best.

  • Setting priorities is invaluable. Prioritization is essential because there are so many problems that a PM must deal with every day. My favorite quotation on this is from Rakuten user Yee Jie. Viki, A competent product manager extinguishes fires. A good product manager lets things burn and then prioritizes.

This summary isn't enough to capture what excellent PMs claim it requires. Read below!

What qualities make a successful product manager?

Themed quotes are alphabetized by author.

Embrace your user/customer

Aeriel Dela Paz, Rainmaking Venture Architect, ex-GCash Product Head

Great PMs know what customers need even when they don’t say it directly. It’s about reading between the lines and going through the numbers to address that need.

Anders Nordahl, OrkestraSCS's Product Manager

Understanding the vision of your customer is as important as to get the customer to buy your vision

Angel Mendoza, MetaverseGo's Product Head

Most people think that to be a great product manager, you must have technical know-how. It’s textbook and I do think it is helpful to some extent, but for me the secret sauce is EMPATHY — the ability to see and feel things from someone else’s perspective. You can’t create a solution without deeply understanding the problem.

Senior Product Manager, Tokopedia

Focus on delivering value and helping people (consumer as well as colleague) and everything else will follow

Darren Lau, Deloitte Digital's Head of Customer Experience

Start with the users, and work backwards. Don’t have a solution looking for a problem

Darryl Tan, Grab Product Manager

I would say that a great product manager is able to identify the crucial problems to solve through strong user empathy and synthesis of insights

Diego Perdana, Kitalulus Senior Product Manager

I think to be a great product manager you need to be obsessed with customer problems and most important is solve the right problem with the right solution

Senior Product Manager, AirAsia

Lot of common sense + Customer Obsession. The most important role of a Product manager is to bring clarity of a solution. Your product is good if it solves customer problems. Your product is great if it solves an eco-system problem and disrupts the business in a positive way.

Edward Xie, Mastercard Managing Consultant, ex-Shopee Product Manager

Perfect your product, but be prepared to compromise for right users

AVP Product, Shipper

For me, a great product manager need to be rational enough to find the business opportunities while obsessing the customers.

Janani Gopalakrishnan is a senior product manager of a stealth firm.

While as a good PM it’s important to be data-driven, to be a great PM one needs to understand that their empathy for their users’ emotions must exceed all logic and data. Great PMs also make these product discussions thrive within the team by intently listening to all the members thoughts and influence the team’s skin in the game positively.

Director, Product Management, Indeed

Great product managers put their users first. They discover problems that matter most to their users and inspire their team to find creative solutions.

Grab's Senior Product Manager Lakshay Kalra

Product management is all about finding and solving most important user problems

Quipper's Mega Puji Saraswati

First of all, always remember the value of “user first” to solve what user really needs (the main problem) for guidance to arrange the task priority and develop new ideas. Second, ownership. Treat the product as your “2nd baby”, and the team as your “2nd family”. Third, maintain a good communication, both horizontally and vertically. But on top of those, always remember to have a work — life balance, and know exactly the priority in life :)

Senior Product Manager, Prosa.AI Miswanto Miswanto

A great Product Manager is someone who can be the link between customer needs with the readiness and flexibility of the team. So that it can provide, build, and produce a product that is useful and helps the community to carry out their daily activities. And He/She can improve product quality ongoing basis or continuous to help provide solutions for users or our customer.

Lead Product Manager, Tokopedia, Oriza Wahyu Utami

Be a great listener, be curious and be determined. every great product manager have the ability to listen the pain points and understand the problems, they are always curious on the users feedback, and they also very determined to look for the solutions that benefited users and the business.

99 Group CPO Rajesh Sangati

The advantage of focusing on customers: it’s impossible to overshoot

Ray Jang, founder of Scenius, formerly of ByteDance

The difference between good and great product managers is that great product managers are willing to go the unsexy and unglamorous extra mile by rolling up their sleeves and ironing out all minutiae details of the product such that when the user uses the product, they can’t help but say “This was made for me.”

BCG Digital Ventures' Sid Narayanan

Great product managers ensure that what gets built and shipped is at the intersection of what creates value for the customer and for the business that’s building the product…often times, especially in today’s highly liquid funding environment, the unit economics, aka ensuring that what gets shipped creates value for the business and is sustainable, gets overlooked

Stephanie Brownlee, BCG Digital Ventures Product Manager

There is software in the world that does more harm than good to people and society. Great Product Managers build products that solve problems not create problems

Experiment constantly

Delivery Hero's Abhishek Muralidharan

Embracing your failure is the key to become a great Product Manager

DeliveryHero's Anuraag Burman

Product Managers should be thick skinned to deal with criticism and the stomach to take risk and face failures.

DataSpark Product Head Apurva Lawale

Great product managers enjoy the creative process with their team to deliver intuitive user experiences to benefit users.

Dexter Zhuang, Xendit Product Manager

The key to creating winning products is building what customers want as quickly as you can — testing and learning along the way.

PayPal's Jay Ko

To me, great product managers always remain relentlessly curious. They are empathetic leaders and problem solvers that glean customer insights into building impactful products

Home Credit Philippines' Jedd Flores

Great Product Managers are the best dreamers; they think of what can be possible for the customers, for the company and the positive impact that it will have in the industry that they’re part of

Set priorities first, foremost, foremost.

HBO Go Product Manager Akshay Ishwar

Good product managers strive to balance the signal to noise ratio, Great product managers know when to turn the dials for each up exactly

Zuellig Pharma's Guojie Su

Have the courage to say no. Managing egos and request is never easy and rejecting them makes it harder but necessary to deliver the best value for the customers.

Ninja Van's John Prawira

(1) PMs should be able to ruthlessly prioritize. In order to be effective, PMs should anchor their product development process with their north stars (success metrics) and always communicate with a purpose. (2) User-first when validating assumptions. PMs should validate assumptions early and often to manage risk when leading initiatives with a focus on generating the highest impact to solving a particular user pain-point. We can’t expect a product/feature launch to be perfect (there might be bugs or we might not achieve our success metric — which is where iteration comes in), but we should try our best to optimize on user-experience earlier on.

Nium Product Manager Keika Sugiyama

I’d say a great PM holds the ability to balance ruthlessness and empathy at the same time. It’s easier said than done for sure!

ShopBack product manager Li Cai

Great product managers are like great Directors of movies. They do not create great products/movies by themselves. They deliver it by Defining, Prioritising, Energising the team to deliver what customers love.

Quincus' Michael Lim

A great product manager, keeps a pulse on the company’s big picture, identifies key problems, and discerns its rightful prioritization, is able to switch between the macro perspective to micro specifics, and communicates concisely with humility that influences naturally for execution

Mathieu François-Barseghian, SVP, Citi Ventures

“You ship your org chart”. This is Conway’s Law short version (1967!): the fundamental socio-technical driver behind innovation successes (Netflix) and failures (your typical bank). The hype behind micro-services is just another reflection of Conway’s Law

Mastercard's Regional Product Manager Nikhil Moorthy

A great PM should always look to build products which are scalable & viable , always keep the end consumer journey in mind. Keeping things simple & having a MVP based approach helps roll out products faster. One has to test & learn & then accordingly enhance / adapt, these are key to success

Rendy Andi, Tokopedia Product Manager

Articulate a clear vision and the path to get there, Create a process that delivers the best results and Be serious about customers.

Senior Product Manager, DANA Indonesia

Own the problem, not the solution — Great PMs are outstanding problem preventers. Great PMs are discerning about which problems to prevent, which problems to solve, and which problems not to solve

Tat Leong Seah, LionsBot International Senior UX Engineer, ex-ViSenze Product Manager

Prioritize outcomes for your users, not outputs of your system” or more succinctly “be agile in delivering value; not features”

Senior Product Manager, Rakuten Viki

A good product manager puts out fires. A great product manager lets fires burn and prioritize from there

acquire fundamental soft skills

Oracle NetSuite's Astrid April Dominguez

Personally, i believe that it takes grit, empathy, and optimistic mindset to become a great PM

Ovo Lead Product Manager Boy Al Idrus

Contrary to popular beliefs, being a great product manager doesn’t have anything to do with technicals, it sure plays a part but most important weapons are: understanding pain points of users, project management, sympathy in leadership and business critical skills; these 4 aspects would definitely help you to become a great product manager.

PwC Product Manager Eric Koh

Product managers need to be courageous to be successful. Courage is required to dive deep, solving big problems at its root and also to think far and dream big to achieve bold visions for your product

Ninja Van's Product Director

In my opinion the two most important ingredients to become a successful product manager is: 1. Strong critical thinking 2. Strong passion for the work. As product managers, we typically need to solve very complex problems where the answers are often very ambiguous. The work is tough and at times can be really frustrating. The 2 ingredients I mentioned earlier will be critical towards helping you to slowly discover the solution that may become a game changer.

PayPal's Lead Product Manager

A great PM has an eye of a designer, the brain of an engineer and the tongue of a diplomat

Product Manager Irene Chan

A great Product Manager is able to think like a CEO of the company. Visionary with Agile Execution in mind

Isabella Yamin, Rakuten Viki Product Manager

There is no one model of being a great product person but what I’ve observed from people I’ve had the privilege working with is an overflowing passion for the user problem, sprinkled with a knack for data and negotiation

Google product manager Jachin Cheng

Great product managers start with abundant intellectual curiosity and grow into a classic T-shape. Horizontally: generalists who range widely, communicate fluidly and collaborate easily cross-functionally, connect unexpected dots, and have the pulse both internally and externally across users, stakeholders, and ecosystem players. Vertically: deep product craftsmanship comes from connecting relentless user obsession with storytelling, business strategy with detailed features and execution, inspiring leadership with risk mitigation, and applying the most relevant tools to solving the right problems.

Jene Lim, Experian's Product Manager

3 Cs and 3 Rs. Critical thinking , Customer empathy, Creativity. Resourcefulness, Resilience, Results orientation.

Nirenj George, Envision Digital's Security Product Manager

A great product manager is someone who can lead, collaborate and influence different stakeholders around the product vision, and should be able to execute the product strategy based on customer insights, as well as take ownership of the product roadmap to create a greater impact on customers.

Grab's Lead Product Manager

Product Management is a multi-dimensional role that looks very different across each product team so each product manager has different challenges to deal with but what I have found common among great product managers is ability to create leverage through their efforts to drive outsized impacts for their products. This leverage is built using data with intuition, building consensus with stakeholders, empowering their teams and focussed efforts on needle moving work.

NCS Product Manager Umar Masagos

To be a great product manager, one must master both the science and art of Product Management. On one hand, you need have a strong understanding of the tools, metrics and data you need to drive your product. On the other hand, you need an in-depth understanding of your organization, your target market and target users, which is often the more challenging aspect to master.

M1 product manager Wei Jiao Keong

A great product manager is multi-faceted. First, you need to have the ability to see the bigger picture, yet have a keen eye for detail. Secondly, you are empathetic and is able to deliver products with exceptional user experience while being analytical enough to achieve business outcomes. Lastly, you are highly resourceful and independent yet comfortable working cross-functionally.

Yudha Utomo, ex-Senior Product Manager, Tokopedia

A great Product Manager is essentially an effective note-taker. In order to achieve the product goals, It is PM’s job to ensure objective has been clearly conveyed, efforts are assessed, and tasks are properly tracked and managed. PM can do this by having top-notch documentation skills.

Will Lockett

Will Lockett

3 years ago

Tesla recently disclosed its greatest secret.

Photo by Taun Stewart on Unsplash

The VP has revealed a secret that should frighten the rest of the EV world.

Tesla led the EV revolution. Elon Musk's invention offers a viable alternative to gas-guzzlers. Tesla has lost ground in recent years. VW, BMW, Mercedes, and Ford offer EVs with similar ranges, charging speeds, performance, and cost. Tesla's next-generation 4680 battery pack, Roadster, Cybertruck, and Semi were all delayed. CATL offers superior batteries than the 4680. Martin Viecha, Tesla's Vice President, recently told Business Insider something that startled the EV world and will establish Tesla as the EV king.

Viecha mentioned that Tesla's production costs have dropped 57% since 2017. This isn't due to cheaper batteries or devices like Model 3. No, this is due to amazing factory efficiency gains.

Musk wasn't crazy to want a nearly 100% automated production line, and Tesla's strategy of sticking with one model and improving it has paid off. Others change models every several years. This implies they must spend on new R&D, set up factories, and modernize service and parts systems. All of this costs a ton of money and prevents them from refining production to cut expenses.

Meanwhile, Tesla updates its vehicles progressively. Everything from the backseats to the screen has been enhanced in a 2022 Model 3. Tesla can refine, standardize, and cheaply produce every part without changing the production line.

In 2017, Tesla's automobile production averaged $84,000. In 2022, it'll be $36,000.

Mr. Viecha also claimed that new factories in Shanghai and Berlin will be significantly cheaper to operate once fully operating.

Tesla's hand is visible. Tesla selling $36,000 cars for $60,000 This barely beats the competition. Model Y long-range costs just over $60,000. Tesla makes $24,000+ every sale, giving it a 40% profit margin, one of the best in the auto business.

VW I.D4 costs about the same but makes no profit. Tesla's rivals face similar challenges. Their EVs make little or no profit.

Tesla costs the same as other EVs, but they're in a different league.

But don't forget that the battery pack accounts for 40% of an EV's cost. Tesla may soon fully utilize its 4680 battery pack.

The 4680 battery pack has larger cells and a unique internal design. This means fewer cells are needed for a car, making it cheaper to assemble and produce (per kWh). Energy density and charge speeds increase slightly.

Tesla underestimated the difficulty of making this revolutionary new cell. Each time they try to scale up production, quality drops and rejected cells rise.

Tesla recently installed this battery pack in Model Ys and is scaling production. If they succeed, Tesla battery prices will plummet.

Tesla's Model Ys 2170 battery costs $11,000. The same size pack with 4680 cells costs $3,400 less. Once scaled, it could be $5,500 (50%) less. The 4680 battery pack could reduce Tesla production costs by 20%.

With these cost savings, Tesla could sell Model Ys for $40,000 while still making a profit. They could offer a $25,000 car.

Even with new battery technology, it seems like other manufacturers will struggle to make EVs profitable.

Teslas cost about the same as competitors, so don't be fooled. Behind the scenes, they're still years ahead, and the 4680 battery pack and new factories will only increase that lead. Musk faces a first. He could sell Teslas at current prices and make billions while other manufacturers struggle. Or, he could massively undercut everyone and crush the competition once and for all. Tesla and Elon win.

Joe Procopio

Joe Procopio

3 years ago

Provide a product roadmap that can withstand startup velocities

This is how to build a car while driving.

Building a high-growth startup is compared to building a car while it's speeding down the highway.

How to plan without going crazy? Or, without losing team, board, and investor buy-in?

I just delivered our company's product roadmap for the rest of the year. Complete. Thorough. Page-long. I'm optimistic about its chances of surviving as everything around us changes, from internal priorities to the global economy.

It's tricky. This isn't the first time I've created a startup roadmap. I didn't invent a document. It took time to deliver a document that will be relevant for months.

Goals matter.

Although they never change, goals are rarely understood.

This is the third in a series about a startup's unique roadmapping needs. Velocity is the intensity at which a startup must produce to survive.

A high-growth startup moves at breakneck speed, which I alluded to when I said priorities and economic factors can change daily or weekly.

At that speed, a startup's roadmap must be flexible, bend but not break, and be brief and to the point. I can't tell you how many startups and large companies develop a product roadmap every quarter and then tuck it away.

Big, wealthy companies can do this. It's suicide for a startup.

The drawer thing happens because startup product roadmaps are often valid for a short time. The roadmap is a random list of features prioritized by different company factions and unrelated to company goals.

It's not because the goals changed that a roadmap is shelved or ignored. Because the company's goals were never communicated or documented in the context of its product.

In the previous post, I discussed how to turn company goals into a product roadmap. In this post, I'll show you how to make a one-page startup roadmap.

In a future post, I'll show you how to follow this roadmap. This roadmap helps you track company goals, something a roadmap must do.

Be vague for growth, but direct for execution.

Here's my plan. The real one has more entries and more content in each.

You can open this as an image at 1920 pixels

Let's discuss smaller boxes.

Product developers and engineers know that the further out they predict, the more wrong they'll be. When developing the product roadmap, this rule is ignored. Then it bites us three, six, or nine months later when we haven't even started.

Why do we put everything in a product roadmap like a project plan?

Yes, I know. We use it when the product roadmap isn't goal-based.

A goal-based roadmap begins with a document that outlines each goal's idea, execution, growth, and refinement.

You can open this as an image at 960 pixels

Once the goals are broken down into epics, initiatives, projects, and programs, only the idea and execution phases should be modeled. Any goal growth or refinement items should be vague and loosely mapped.

Why? First, any idea or execution-phase goal will result in growth initiatives that are unimaginable today. Second, internal priorities and external factors will change, but the goals won't. Locking items into calendar slots reduces flexibility and forces deviation from the single source of truth.

No soothsayers. Predicting the future is pointless; just prepare.

A map is useless if you don't know where you're going.

As we speed down the road, the car and the road will change. Goals define the destination.

This quarter and next quarter's roadmap should be set. After that, you should track destination milestones, not how to get there.

When you do that, even the most critical investors will understand the roadmap and buy in. When you track progress at the end of the quarter and revise your roadmap, the destination won't change.

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DC Palter

DC Palter

2 years ago

Why Are There So Few Startups in Japan?

Japan's startup challenge: 7 reasons

Photo by Timo Volz on Unsplash

Every day, another Silicon Valley business is bought for a billion dollars, making its founders rich while growing the economy and improving consumers' lives.

Google, Amazon, Twitter, and Medium dominate our daily lives. Tesla automobiles and Moderna Covid vaccinations.

The startup movement started in Silicon Valley, California, but the rest of the world is catching up. Global startup buzz is rising. Except Japan.

644 of CB Insights' 1170 unicorns—successful firms valued at over $1 billion—are US-based. China follows with 302 and India third with 108.

Japan? 6!

1% of US startups succeed. The third-largest economy is tied with small Switzerland for startup success.

Mexico (8), Indonesia (12), and Brazil (12) have more successful startups than Japan (16). South Korea has 16. Yikes! Problem?

Why Don't Startups Exist in Japan More?

Not about money. Japanese firms invest in startups. To invest in startups, big Japanese firms create Silicon Valley offices instead of Tokyo.

Startups aren't the issue either. Local governments are competing to be Japan's Shirikon Tani, providing entrepreneurs financing, office space, and founder visas.

Startup accelerators like Plug and Play in Tokyo, Osaka, and Kyoto, the Startup Hub in Kobe, and Google for Startups are many.

Most of the companies I've encountered in Japan are either local offices of foreign firms aiming to expand into the Japanese market or small businesses offering local services rather than disrupting a staid industry with new ideas.

There must be a reason Japan can develop world-beating giant corporations like Toyota, Nintendo, Shiseido, and Suntory but not inventive startups.

Culture, obviously. Japanese culture excels in teamwork, craftsmanship, and quality, but it hates moving fast, making mistakes, and breaking things.

If you have a brilliant idea in Silicon Valley, quit your job, get money from friends and family, and build a prototype. To fund the business, you approach angel investors and VCs.

Most non-startup folks don't aware that venture capitalists don't want good, profitable enterprises. That's wonderful if you're developing a solid small business to consult, open shops, or make a specialty product. However, you must pay for it or borrow money. Venture capitalists want moon rockets. Silicon Valley is big or bust. Almost 90% will explode and crash. The few successes are remarkable enough to make up for the failures.

Silicon Valley's high-risk, high-reward attitude contrasts with Japan's incrementalism. Japan makes the best automobiles and cleanrooms, but it fails to produce new items that grow the economy.

Changeable? Absolutely. But, what makes huge manufacturing enterprises successful and what makes Japan a safe and comfortable place to live are inextricably connected with the lack of startups.

Barriers to Startup Development in Japan

These are the 7 biggest obstacles to Japanese startup success.

  1. Unresponsive Employment Market

While the lifelong employment system in Japan is evolving, the average employee stays at their firm for 12 years (15 years for men at large organizations) compared to 4.3 years in the US. Seniority, not experience or aptitude, determines career routes, making it tough to quit a job to join a startup and then return to corporate work if it fails.

  1. Conservative Buyers

Even if your product is buggy and undocumented, US customers will migrate to a cheaper, superior one. Japanese corporations demand perfection from their trusted suppliers and keep with them forever. Startups need income fast, yet product evaluation takes forever.

  1. Failure intolerance

Japanese business failures harm lives. Failed forever. It hinders risk-taking. Silicon Valley embraces failure. Build another startup if your first fails. Build a third if that fails. Every setback is viewed as a learning opportunity for success.

4. No Corporate Purchases

Silicon Valley industrial giants will buy fast-growing startups for a lot of money. Many huge firms have stopped developing new goods and instead buy startups after the product is validated.

Japanese companies prefer in-house product development over startup acquisitions. No acquisitions mean no startup investment and no investor reward.

Startup investments can also be monetized through stock market listings. Public stock listings in Japan are risky because the Nikkei was stagnant for 35 years while the S&P rose 14x.

5. Social Unity Above Wealth

In Silicon Valley, everyone wants to be rich. That creates a competitive environment where everyone wants to succeed, but it also promotes fraud and societal problems.

Japan values communal harmony above individual success. Wealthy folks and overachievers are avoided. In Japan, renegades are nearly impossible.

6. Rote Learning Education System

Japanese high school graduates outperform most Americans. Nonetheless, Japanese education is known for its rote memorization. The American system, which fails too many kids, emphasizes creativity to create new products.

  1. Immigration.

Immigrants start 55% of successful Silicon Valley firms. Some come for university, some to escape poverty and war, and some are recruited by Silicon Valley startups and stay to start their own.

Japan is difficult for immigrants to start a business due to language barriers, visa restrictions, and social isolation.

How Japan Can Promote Innovation

Patchwork solutions to deep-rooted cultural issues will not work. If customers don't buy things, immigration visas won't aid startups. Startups must have a chance of being acquired for a huge sum to attract investors. If risky startups fail, employees won't join.

Will Japan never have a startup culture?

Once a consensus is reached, Japan changes rapidly. A dwindling population and standard of living may lead to such consensus.

Toyota and Sony were firms with renowned founders who used technology to transform the world. Repeatable.

Silicon Valley is flawed too. Many people struggle due to wealth disparities, job churn and layoffs, and the tremendous ups and downs of the economy caused by stock market fluctuations.

The founders of the 10% successful startups are heroes. The 90% that fail and return to good-paying jobs with benefits are never mentioned.

Silicon Valley startup culture and Japanese corporate culture are opposites. Each have pros and cons. Big Japanese corporations make the most reliable, dependable, high-quality products yet move too slowly. That's good for creating cars, not social networking apps.

Can innovation and success be encouraged without eroding social cohesion? That can motivate software firms to move fast and break things while recognizing the beauty and precision of expert craftsmen? A hybrid culture where Japan can make the world's best and most original items. Hopefully.

Vitalik

Vitalik

4 years ago

An approximate introduction to how zk-SNARKs are possible (part 2)

If tasked with the problem of coming up with a zk-SNARK protocol, many people would make their way to this point and then get stuck and give up. How can a verifier possibly check every single piece of the computation, without looking at each piece of the computation individually? But it turns out that there is a clever solution.

Polynomials

Polynomials are a special class of algebraic expressions of the form:

  • x+5
  • x^4
  • x^3+3x^2+3x+1
  • 628x^{271}+318x^{270}+530x^{269}+…+69x+381

i.e. they are a sum of any (finite!) number of terms of the form cx^k

There are many things that are fascinating about polynomials. But here we are going to zoom in on a particular one: polynomials are a single mathematical object that can contain an unbounded amount of information (think of them as a list of integers and this is obvious). The fourth example above contained 816 digits of tau, and one can easily imagine a polynomial that contains far more.

Furthermore, a single equation between polynomials can represent an unbounded number of equations between numbers. For example, consider the equation A(x)+ B(x) = C(x). If this equation is true, then it's also true that:

  • A(0)+B(0)=C(0)
  • A(1)+B(1)=C(1)
  • A(2)+B(2)=C(2)
  • A(3)+B(3)=C(3)

And so on for every possible coordinate. You can even construct polynomials to deliberately represent sets of numbers so you can check many equations all at once. For example, suppose that you wanted to check:

  • 12+1=13
  • 10+8=18
  • 15+8=23
  • 15+13=28

You can use a procedure called Lagrange interpolation to construct polynomials A(x) that give (12,10,15,15) as outputs at some specific set of coordinates (eg. (0,1,2,3)), B(x) the outputs (1,8,8,13) on thos same coordinates, and so forth. In fact, here are the polynomials:

  • A(x)=-2x^3+\frac{19}{2}x^2-\frac{19}{2}x+12
  • B(x)=2x^3-\frac{19}{2}x^2+\frac{29}{2}x+1
  • C(x)=5x+13

Checking the equation A(x)+B(x)=C(x) with these polynomials checks all four above equations at the same time.

Comparing a polynomial to itself

You can even check relationships between a large number of adjacent evaluations of the same polynomial using a simple polynomial equation. This is slightly more advanced. Suppose that you want to check that, for a given polynomial F, F(x+2)=F(x)+F(x+1) with the integer range {0,1…89} (so if you also check F(0)=F(1)=1, then F(100) would be the 100th Fibonacci number)

As polynomials, F(x+2)-F(x+1)-F(x) would not be exactly zero, as it could give arbitrary answers outside the range x={0,1…98}. But we can do something clever. In general, there is a rule that if a polynomial P is zero across some set S=\{x_1,x_2…x_n\} then it can be expressed as P(x)=Z(x)*H(x), where Z(x)=(x-x_1)*(x-x_2)*…*(x-x_n) and H(x) is also a polynomial. In other words, any polynomial that equals zero across some set is a (polynomial) multiple of the simplest (lowest-degree) polynomial that equals zero across that same set.

Why is this the case? It is a nice corollary of polynomial long division: the factor theorem. We know that, when dividing P(x) by Z(x), we will get a quotient Q(x) and a remainder R(x) is strictly less than that of Z(x). Since we know that P is zero on all of S, it means that R has to be zero on all of S as well. So we can simply compute R(x) via polynomial interpolation, since it's a polynomial of degree at most n-1 and we know n values (the zeros at S). Interpolating a polynomial with all zeroes gives the zero polynomial, thus R(x)=0 and H(x)=Q(x).

Going back to our example, if we have a polynomial F that encodes Fibonacci numbers (so F(x+2)=F(x)+F(x+1) across x=\{0,1…98\}), then I can convince you that F actually satisfies this condition by proving that the polynomial P(x)=F(x+2)-F(x+1)-F(x) is zero over that range, by giving you the quotient:
H(x)=\frac{F(x+2)-F(x+1)-F(x)}{Z(x)}
Where Z(x) = (x-0)*(x-1)*…*(x-98).
You can calculate Z(x) yourself (ideally you would have it precomputed), check the equation, and if the check passes then F(x) satisfies the condition!

Now, step back and notice what we did here. We converted a 100-step-long computation into a single equation with polynomials. Of course, proving the N'th Fibonacci number is not an especially useful task, especially since Fibonacci numbers have a closed form. But you can use exactly the same basic technique, just with some extra polynomials and some more complicated equations, to encode arbitrary computations with an arbitrarily large number of steps.

see part 3

Tora Northman

Tora Northman

3 years ago

Pixelmon NFTs are so bad, they are almost good!

Bored Apes prices continue to rise, HAPEBEAST launches, Invisible Friends hype continues to grow. Sadly, not all projects are as successful.
Of course, there are many factors to consider when buying an NFT. Is the project a scam? Will the reveal derail the project? Possibly, but when Pixelmon first teased its launch, it generated a lot of buzz.

With a primary sale mint price of 3 ETH ($8,100 USD), it started as an expensive project, with plenty of fans willing to invest in what was sold as a game. After it was revealed, it fell rapidly.
Why? It was overpromised and under delivered.

According to the project's creator[^1], the funds generated will be used to develop the artwork. "The Pixelmon reveal was wrong. This is what our Pixelmon look like in-game. "Despite the fud, I will not go anywhere," he wrote on Twitter. The goal remains. The funds will still be used to build our game. I will finish this project."

The project raised $70 million USD, but the NFTs buyers received were not the project's original teasers. Some call it "the worst NFT project ever," while others call it a complete scam.

But there's hope for some buyers. Kevin emerged from the ashes as the project was roasted over the fire.

A Minecraft character meets Salad Fingers - that's Kevin. He's a frog-like creature whose reveal was such a terrible NFT that it became part of history – and a meme.

If you're laughing at people paying $8K for a silly pixelated image, you might need to take it back. Precisely because of this, lucky holders who minted Kevin have been able to sell the now-memed NFT for over 8 ETH (around $24,000 USD), with some currently listed for 100 ETH.

Of course, Twitter has been awash in memes mocking those who invested in the project, because what else can you do when so many people lose money?

It's still unclear if the NFT project is a scam, but the team behind it was hired on Upwork. There's still hope for redemption, but Kevin's rise to fame appears to be the only positive outcome so far.

[^1] This is not the first time the creator (A 20-yo New Zealanders) has sought money via an online platform and had people claiming he under-delivered.  He raised $74,000 on Kickstarter for a card game called Psycho Chicken. There are hundreds of comments on the Kickstarter project saying they haven't received the product and pleading for a refund or an update.