More on Society & Culture

Tim Smedley
2 years ago
When Investment in New Energy Surpassed That in Fossil Fuels (Forever)
A worldwide energy crisis might have hampered renewable energy and clean tech investment. Nope.
BNEF's 2023 Energy Transition Investment Trends study surprised and encouraged. Global energy transition investment reached $1 trillion for the first time ($1.11t), up 31% from 2021. From 2013, the clean energy transition has come and cannot be reversed.
BNEF Head of Global Analysis Albert Cheung said our findings ended the energy crisis's influence on renewable energy deployment. Energy transition investment has reached a record as countries and corporations implement transition strategies. Clean energy investments will soon surpass fossil fuel investments.
The table below indicates the tripping point, which means the energy shift is occuring today.
BNEF calls money invested on clean technology including electric vehicles, heat pumps, hydrogen, and carbon capture energy transition investment. In 2022, electrified heat received $64b and energy storage $15.7b.
Nonetheless, $495b in renewables (up 17%) and $466b in electrified transport (up 54%) account for most of the investment. Hydrogen and carbon capture are tiny despite the fanfare. Hydrogen received the least funding in 2022 at $1.1 billion (0.1%).
China dominates investment. China spends $546 billion on energy transition, half the global amount. Second, the US total of $141 billion in 2022 was up 11% from 2021. With $180 billion, the EU is unofficially second. China invested 91% in battery technologies.
The 2022 transition tipping point is encouraging, but the BNEF research shows how far we must go to get Net Zero. Energy transition investment must average $4.55 trillion between 2023 and 2030—three times the amount spent in 2022—to reach global Net Zero. Investment must be seven times today's record to reach Net Zero by 2050.
BNEF 2023 Energy Transition Investment Trends.
As shown in the graph above, BNEF experts have been using their crystal balls to determine where that investment should go. CCS and hydrogen are still modest components of the picture. Interestingly, they see nuclear almost fading. Active transport advocates like me may have something to say about the massive $4b in electrified transport. If we focus on walkable 15-minute cities, we may need fewer electric automobiles. Though we need more electric trains and buses.
Albert Cheung of BNEF emphasizes the challenge. This week's figures promise short-term job creation and medium-term energy security, but more investment is needed to reach net zero in the long run.
I expect the BNEF Energy Transition Investment Trends report to show clean tech investment outpacing fossil fuels investment every year. Finally saying that is amazing. It's insufficient. The planet must maintain its electric (not gas) pedal. In response to the research, Christina Karapataki, VC at Breakthrough Energy Ventures, a clean tech investment firm, tweeted: Clean energy investment needs to average more than 3x this level, for the remainder of this decade, to get on track for BNEFs Net Zero Scenario. Go!

Liz Martin
3 years ago
What Motivated Amazon to Spend $1 Billion for The Rings of Power?
Amazon's Rings of Power is the most costly TV series ever made. This is merely a down payment towards Amazon's grand goal.
Here's a video:
Amazon bought J.R.R. Tolkien's fantasy novels for $250 million in 2017. This agreement allows Amazon to create a Tolkien series for Prime Video.
The business spent years developing and constructing a Lord of the Rings prequel. Rings of Power premiered on September 2, 2022.
It drew 25 million global viewers in 24 hours. Prime Video's biggest debut.
An Exorbitant Budget
The most expensive. First season cost $750 million to $1 billion, making it the most costly TV show ever.
Jeff Bezos has spent years looking for the next Game of Thrones, a critically and commercially successful original series. Rings of Power could help.
Why would Amazon bet $1 billion on one series?
It's Not Just About the Streaming War
It's simple to assume Amazon just wants to win. Since 2018, the corporation has been fighting Hulu, Netflix, HBO, Apple, Disney, and NBC. Each wants your money, talent, and attention. Amazon's investment goes beyond rivalry.
Subscriptions Are the Bait
Audible, Amazon Music, and Prime Video are subscription services, although the company's fundamental business is retail. Amazon's online stores contribute over 50% of company revenue. Subscription services contribute 6.8%. The company's master plan depends on these subscriptions.
Streaming videos on Prime increases membership renewals. Free trial participants are more likely to join. Members buy twice as much as non-members.
Amazon Studios doesn't generate original programming to earn from Prime Video subscriptions. It aims to retain and attract clients.
Amazon can track what you watch and buy. Its algorithm recommends items and services. Mckinsey says you'll use more Amazon products, shop at Amazon stores, and watch Amazon entertainment.
In 2015, the firm launched the first season of The Man in the High Castle, a dystopian alternate history TV series depicting a world ruled by Nazi Germany and Japan after World War II.
This $72 million production earned two Emmys. It garnered 1.15 million new Prime users globally.
When asked about his Hollywood investment, Bezos said, "A Golden Globe helps us sell more shoes."
Selling more footwear
Amazon secured a deal with DirecTV to air Thursday Night Football in restaurants and bars. First streaming service to have exclusive NFL games.
This isn't just about Thursday night football, says media analyst Ritchie Greenfield. This sells t-shirts. This may be a ticket. Amazon does more than stream games.
The Rings of Power isn't merely a production showcase, either. This sells Tolkien's fantasy novels such Lord of the Rings, The Hobbit, and The Silmarillion.
This tiny commitment keeps you in Amazon's ecosystem.

Logan Rane
2 years ago
I questioned Chat-GPT for advice on the top nonfiction books. Here's What It Suggests
You have to use it.
Chat-GPT is a revolution.
All social media outlets are discussing it. How it will impact the future and different things.
True.
I've been using Chat-GPT for a few days, and it's a rare revolution. It's amazing and will only improve.
I asked Chat-GPT about the best non-fiction books. It advised this, albeit results rely on interests.
The Immortal Life of Henrietta Lacks
by Rebecca Skloot
Science, Biography
A impoverished tobacco farmer dies of cervical cancer in The Immortal Life of Henrietta Lacks. Her cell strand helped scientists treat polio and other ailments.
Rebecca Skloot discovers about Henrietta, her family, how the medical business exploited black Americans, and how her cells can live forever in a fascinating and surprising research.
You ought to read it.
if you want to discover more about the past of medicine.
if you want to discover more about American history.
Bad Blood: Secrets and Lies in a Silicon Valley Startup
by John Carreyrou
Tech, Bio
Bad Blood tells the terrifying story of how a Silicon Valley tech startup's blood-testing device placed millions of lives at risk.
John Carreyrou, a Pulitzer Prize-winning journalist, wrote this book.
Theranos and its wunderkind CEO, Elizabeth Holmes, climbed to popularity swiftly and then plummeted.
You ought to read it.
if you are a start-up employee.
specialists in medicine.
The Power of Now: A Guide to Spiritual Enlightenment
by Eckhart Tolle
Self-improvement, Spirituality
The Power of Now shows how to stop suffering and attain inner peace by focusing on the now and ignoring your mind.
The book also helps you get rid of your ego, which tries to control your ideas and actions.
If you do this, you may embrace the present, reduce discomfort, strengthen relationships, and live a better life.
You ought to read it.
if you're looking for serenity and illumination.
If you believe that you are ruining your life, stop.
if you're not happy.
The 7 Habits of Highly Effective People
by Stephen R. Covey
Profession, Success
The 7 Habits of Highly Effective People is an iconic self-help book.
This vital book offers practical guidance for personal and professional success.
This non-fiction book is one of the most popular ever.
You ought to read it.
if you want to reach your full potential.
if you want to discover how to achieve all your objectives.
if you are just beginning your journey toward personal improvement.
Sapiens: A Brief History of Humankind
by Yuval Noah Harari
Science, History
Sapiens explains how our species has evolved from our earliest ancestors to the technology age.
How did we, a species of hairless apes without tails, come to control the whole planet?
It describes the shifts that propelled Homo sapiens to the top.
You ought to read it.
if you're interested in discovering our species' past.
if you want to discover more about the origins of human society and culture.
You might also like

Sofien Kaabar, CFA
3 years ago
How to Make a Trading Heatmap
Python Heatmap Technical Indicator
Heatmaps provide an instant overview. They can be used with correlations or to predict reactions or confirm the trend in trading. This article covers RSI heatmap creation.
The Market System
Market regime:
Bullish trend: The market tends to make higher highs, which indicates that the overall trend is upward.
Sideways: The market tends to fluctuate while staying within predetermined zones.
Bearish trend: The market has the propensity to make lower lows, indicating that the overall trend is downward.
Most tools detect the trend, but we cannot predict the next state. The best way to solve this problem is to assume the current state will continue and trade any reactions, preferably in the trend.
If the EURUSD is above its moving average and making higher highs, a trend-following strategy would be to wait for dips before buying and assuming the bullish trend will continue.
Indicator of Relative Strength
J. Welles Wilder Jr. introduced the RSI, a popular and versatile technical indicator. Used as a contrarian indicator to exploit extreme reactions. Calculating the default RSI usually involves these steps:
Determine the difference between the closing prices from the prior ones.
Distinguish between the positive and negative net changes.
Create a smoothed moving average for both the absolute values of the positive net changes and the negative net changes.
Take the difference between the smoothed positive and negative changes. The Relative Strength RS will be the name we use to describe this calculation.
To obtain the RSI, use the normalization formula shown below for each time step.
The 13-period RSI and black GBPUSD hourly values are shown above. RSI bounces near 25 and pauses around 75. Python requires a four-column OHLC array for RSI coding.
import numpy as np
def add_column(data, times):
for i in range(1, times + 1):
new = np.zeros((len(data), 1), dtype = float)
data = np.append(data, new, axis = 1)
return data
def delete_column(data, index, times):
for i in range(1, times + 1):
data = np.delete(data, index, axis = 1)
return data
def delete_row(data, number):
data = data[number:, ]
return data
def ma(data, lookback, close, position):
data = add_column(data, 1)
for i in range(len(data)):
try:
data[i, position] = (data[i - lookback + 1:i + 1, close].mean())
except IndexError:
pass
data = delete_row(data, lookback)
return data
def smoothed_ma(data, alpha, lookback, close, position):
lookback = (2 * lookback) - 1
alpha = alpha / (lookback + 1.0)
beta = 1 - alpha
data = ma(data, lookback, close, position)
data[lookback + 1, position] = (data[lookback + 1, close] * alpha) + (data[lookback, position] * beta)
for i in range(lookback + 2, len(data)):
try:
data[i, position] = (data[i, close] * alpha) + (data[i - 1, position] * beta)
except IndexError:
pass
return data
def rsi(data, lookback, close, position):
data = add_column(data, 5)
for i in range(len(data)):
data[i, position] = data[i, close] - data[i - 1, close]
for i in range(len(data)):
if data[i, position] > 0:
data[i, position + 1] = data[i, position]
elif data[i, position] < 0:
data[i, position + 2] = abs(data[i, position])
data = smoothed_ma(data, 2, lookback, position + 1, position + 3)
data = smoothed_ma(data, 2, lookback, position + 2, position + 4)
data[:, position + 5] = data[:, position + 3] / data[:, position + 4]
data[:, position + 6] = (100 - (100 / (1 + data[:, position + 5])))
data = delete_column(data, position, 6)
data = delete_row(data, lookback)
return dataMake sure to focus on the concepts and not the code. You can find the codes of most of my strategies in my books. The most important thing is to comprehend the techniques and strategies.
My weekly market sentiment report uses complex and simple models to understand the current positioning and predict the future direction of several major markets. Check out the report here:
Using the Heatmap to Find the Trend
RSI trend detection is easy but useless. Bullish and bearish regimes are in effect when the RSI is above or below 50, respectively. Tracing a vertical colored line creates the conditions below. How:
When the RSI is higher than 50, a green vertical line is drawn.
When the RSI is lower than 50, a red vertical line is drawn.
Zooming out yields a basic heatmap, as shown below.
Plot code:
def indicator_plot(data, second_panel, window = 250):
fig, ax = plt.subplots(2, figsize = (10, 5))
sample = data[-window:, ]
for i in range(len(sample)):
ax[0].vlines(x = i, ymin = sample[i, 2], ymax = sample[i, 1], color = 'black', linewidth = 1)
if sample[i, 3] > sample[i, 0]:
ax[0].vlines(x = i, ymin = sample[i, 0], ymax = sample[i, 3], color = 'black', linewidth = 1.5)
if sample[i, 3] < sample[i, 0]:
ax[0].vlines(x = i, ymin = sample[i, 3], ymax = sample[i, 0], color = 'black', linewidth = 1.5)
if sample[i, 3] == sample[i, 0]:
ax[0].vlines(x = i, ymin = sample[i, 3], ymax = sample[i, 0], color = 'black', linewidth = 1.5)
ax[0].grid()
for i in range(len(sample)):
if sample[i, second_panel] > 50:
ax[1].vlines(x = i, ymin = 0, ymax = 100, color = 'green', linewidth = 1.5)
if sample[i, second_panel] < 50:
ax[1].vlines(x = i, ymin = 0, ymax = 100, color = 'red', linewidth = 1.5)
ax[1].grid()
indicator_plot(my_data, 4, window = 500)Call RSI on your OHLC array's fifth column. 4. Adjusting lookback parameters reduces lag and false signals. Other indicators and conditions are possible.
Another suggestion is to develop an RSI Heatmap for Extreme Conditions.
Contrarian indicator RSI. The following rules apply:
Whenever the RSI is approaching the upper values, the color approaches red.
The color tends toward green whenever the RSI is getting close to the lower values.
Zooming out yields a basic heatmap, as shown below.
Plot code:
import matplotlib.pyplot as plt
def indicator_plot(data, second_panel, window = 250):
fig, ax = plt.subplots(2, figsize = (10, 5))
sample = data[-window:, ]
for i in range(len(sample)):
ax[0].vlines(x = i, ymin = sample[i, 2], ymax = sample[i, 1], color = 'black', linewidth = 1)
if sample[i, 3] > sample[i, 0]:
ax[0].vlines(x = i, ymin = sample[i, 0], ymax = sample[i, 3], color = 'black', linewidth = 1.5)
if sample[i, 3] < sample[i, 0]:
ax[0].vlines(x = i, ymin = sample[i, 3], ymax = sample[i, 0], color = 'black', linewidth = 1.5)
if sample[i, 3] == sample[i, 0]:
ax[0].vlines(x = i, ymin = sample[i, 3], ymax = sample[i, 0], color = 'black', linewidth = 1.5)
ax[0].grid()
for i in range(len(sample)):
if sample[i, second_panel] > 90:
ax[1].vlines(x = i, ymin = 0, ymax = 100, color = 'red', linewidth = 1.5)
if sample[i, second_panel] > 80 and sample[i, second_panel] < 90:
ax[1].vlines(x = i, ymin = 0, ymax = 100, color = 'darkred', linewidth = 1.5)
if sample[i, second_panel] > 70 and sample[i, second_panel] < 80:
ax[1].vlines(x = i, ymin = 0, ymax = 100, color = 'maroon', linewidth = 1.5)
if sample[i, second_panel] > 60 and sample[i, second_panel] < 70:
ax[1].vlines(x = i, ymin = 0, ymax = 100, color = 'firebrick', linewidth = 1.5)
if sample[i, second_panel] > 50 and sample[i, second_panel] < 60:
ax[1].vlines(x = i, ymin = 0, ymax = 100, color = 'grey', linewidth = 1.5)
if sample[i, second_panel] > 40 and sample[i, second_panel] < 50:
ax[1].vlines(x = i, ymin = 0, ymax = 100, color = 'grey', linewidth = 1.5)
if sample[i, second_panel] > 30 and sample[i, second_panel] < 40:
ax[1].vlines(x = i, ymin = 0, ymax = 100, color = 'lightgreen', linewidth = 1.5)
if sample[i, second_panel] > 20 and sample[i, second_panel] < 30:
ax[1].vlines(x = i, ymin = 0, ymax = 100, color = 'limegreen', linewidth = 1.5)
if sample[i, second_panel] > 10 and sample[i, second_panel] < 20:
ax[1].vlines(x = i, ymin = 0, ymax = 100, color = 'seagreen', linewidth = 1.5)
if sample[i, second_panel] > 0 and sample[i, second_panel] < 10:
ax[1].vlines(x = i, ymin = 0, ymax = 100, color = 'green', linewidth = 1.5)
ax[1].grid()
indicator_plot(my_data, 4, window = 500)Dark green and red areas indicate imminent bullish and bearish reactions, respectively. RSI around 50 is grey.
Summary
To conclude, my goal is to contribute to objective technical analysis, which promotes more transparent methods and strategies that must be back-tested before implementation.
Technical analysis will lose its reputation as subjective and unscientific.
When you find a trading strategy or technique, follow these steps:
Put emotions aside and adopt a critical mindset.
Test it in the past under conditions and simulations taken from real life.
Try optimizing it and performing a forward test if you find any potential.
Transaction costs and any slippage simulation should always be included in your tests.
Risk management and position sizing should always be considered in your tests.
After checking the above, monitor the strategy because market dynamics may change and make it unprofitable.

nft now
3 years ago
A Guide to VeeFriends and Series 2
VeeFriends is one of the most popular and unique NFT collections. VeeFriends launched around the same time as other PFP NFTs like Bored Ape Yacht Club.
Vaynerchuk (GaryVee) took a unique approach to his large-scale project, which has influenced the NFT ecosystem. GaryVee's VeeFriends is one of the most successful NFT membership use-cases, allowing him to build a community around his creative and business passions.
What is VeeFriends?
GaryVee's NFT collection, VeeFriends, was released on May 11, 2021. VeeFriends [Mini Drops], Book Games, and a forthcoming large-scale "Series 2" collection all stem from the initial drop of 10,255 tokens.
In "Series 1," there are G.O.O. tokens (Gary Originally Owned). GaryVee reserved 1,242 NFTs (over 12% of the supply) for his own collection, so only 9,013 were available at the Series 1 launch.
Each Series 1 token represents one of 268 human traits hand-drawn by Vaynerchuk. Gary Vee's NFTs offer owners incentives.
Who made VeeFriends?
Gary Vaynerchuk, AKA GaryVee, is influential in NFT. Vaynerchuk is the chairman of New York-based communications company VaynerX. Gary Vee, CEO of VaynerMedia, VaynerSports, and bestselling author, is worth $200 million.
GaryVee went from NFT collector to creator, launching VaynerNFT to help celebrities and brands.
Vaynerchuk's influence spans the NFT ecosystem as one of its most prolific voices. He's one of the most influential NFT figures, and his VeeFriends ecosystem keeps growing.
Vaynerchuk, a trend expert, thinks NFTs will be around for the rest of his life and VeeFriends will be a landmark project.
Why use VeeFriends NFTs?
The first VeeFriends collection has sold nearly $160 million via OpenSea. GaryVee insisted that the first 10,255 VeeFriends were just the beginning.
Book Games were announced to the VeeFriends community in August 2021. Mini Drops joined VeeFriends two months later.
Book Games
GaryVee's book "Twelve and a Half: Leveraging the Emotional Ingredients for Business Success" inspired Book Games. Even prior to the announcement Vaynerchuk had mapped out the utility of the book on an NFT scale. Book Games tied his book to the VeeFriends ecosystem and solidified its place in the collection.
GaryVee says Book Games is a layer 2 NFT project with 125,000 burnable tokens. Vaynerchuk's NFT fans were incentivized to buy as many copies of his new book as possible to receive NFT rewards later.
First, a bit about “layer 2.”
Layer 2 blockchain solutions help scale applications by routing transactions away from Ethereum Mainnet (layer 1). These solutions benefit from Mainnet's decentralized security model but increase transaction speed and reduce gas fees.
Polygon (integrated into OpenSea) and Immutable X are popular Ethereum layer 2 solutions. GaryVee chose Immutable X to reduce gas costs (transaction fees). Given the large supply of Book Games tokens, this decision will likely benefit the VeeFriends community, especially if the games run forever.
What's the strategy?
The VeeFriends patriarch announced on Aug. 27, 2021, that for every 12 books ordered during the Book Games promotion, customers would receive one NFT via airdrop. After nearly 100 days, GV sold over a million copies and announced that Book Games would go gamified on Jan. 10, 2022.
Immutable X's trading options make Book Games a "game." Book Games players can trade NFTs for other NFTs, sports cards, VeeCon tickets, and other prizes. Book Games can also whitelist other VeeFirends projects, which we'll cover in Series 2.
VeeFriends Mini Drops
GaryVee launched VeeFriends Mini Drops two months after Book Games, focusing on collaboration, scarcity, and the characters' "cultural longevity."
Spooky Vees, a collection of 31 1/1 Halloween-themed VeeFriends, was released on Halloween. First-come, first-served VeeFriend owners could claim these NFTs.
Mini Drops includes Gift Goat NFTs. By holding the Gift Goat VeeFriends character, collectors will receive 18 exclusive gifts curated by GaryVee and the team. Each gifting experience includes one physical gift and one NFT out of 555, to match the 555 Gift Goat tokens.
Gift Goat holders have gotten NFTs from Danny Cole (Creature World), Isaac "Drift" Wright (Where My Vans Go), Pop Wonder, and more.
GaryVee is poised to release the largest expansion of the VeeFriends and VaynerNFT ecosystem to date with VeeFriends Series 2.
VeeCon 101
By owning VeeFriends NFTs, collectors can join the VeeFriends community and attend VeeCon in 2022. The conference is only open to VeeCon NFT ticket holders (VeeFreinds + possibly more TBA) and will feature Beeple, Steve Aoki, and even Snoop Dogg.
The VeeFreinds floor in 2022 Q1 has remained at 16 ETH ($52,000), making VeeCon unattainable for most NFT enthusiasts. Why would someone spend that much crypto on a Minneapolis "superconference" ticket? Because of Gary Vaynerchuk.
Everything to know about VeeFriends Series 2
Vaynerchuk revealed in April 2022 that the VeeFriends ecosystem will grow by 55,555 NFTs after months of teasing.
With VeeFriends Series 2, each token will cost $995 USD in ETH, allowing NFT enthusiasts to join at a lower cost. The new series will be released on multiple dates in April.
Book Games NFT holders on the Friends List (whitelist) can mint Series 2 NFTs on April 12. Book Games holders have 32,000 NFTs.
VeeFriends Series 1 NFT holders can claim Series 2 NFTs on April 12. This allotment's supply is 10,255, like Series 1's.
On April 25, the public can buy 10,000 Series 2 NFTs. Unminted Friends List NFTs will be sold on this date, so this number may change.
The VeeFriends ecosystem will add 15 new characters (220 tokens each) on April 27. One character will be released per day for 15 days, and the only way to get one is to enter a daily raffle with Book Games tokens.
Series 2 NFTs won't give owners VeeCon access, but they will offer other benefits within the VaynerNFT ecosystem. Book Games and Series 2 will get new token burn mechanics in the upcoming drop.
Visit the VeeFriends blog for the latest collection info.
Where can you buy Gary Vee’s NFTs?
Need a VeeFriend NFT? Gary Vee recommends doing "50 hours of homework" before buying. OpenSea sells VeeFriends NFTs.

Vitalik
3 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
