More on Cooking
Scott Hickmann
3 years ago Draft
This is a draft
My wallpape

Dr. Linda Dahl
3 years ago
We eat corn in almost everything. Is It Important?
Corn Kid got viral on TikTok after being interviewed by Recess Therapy. Tariq, called the Corn Kid, ate a buttery ear of corn in the video. He's corn crazy. He thinks everyone just has to try it. It turns out, whether we know it or not, we already have.
Corn is a fruit, veggie, and grain. It's the second-most-grown crop. Corn makes up 36% of U.S. exports. In the U.S., it's easy to grow and provides high yields, as proven by the vast corn belt spanning the Midwest, Great Plains, and Texas panhandle. Since 1950, the corn crop has doubled to 10 billion bushels.
You say, "Fine." We shouldn't just grow because we can. Why so much corn? What's this corn for?
Why is practical and political. Michael Pollan's The Omnivore's Dilemma has the full narrative. Early 1970s food costs increased. Nixon subsidized maize to feed the public. Monsanto genetically engineered corn seeds to make them hardier, and soon there was plenty of corn. Everyone ate. Woot! Too much corn followed. The powers-that-be had to decide what to do with leftover corn-on-the-cob.
They are fortunate that corn has a wide range of uses.
First, the edible variants. I divide corn into obvious and stealth.
Obvious corn includes popcorn, canned corn, and corn on the cob. This form isn't always digested and often comes out as entire, polka-dotting poop. Cornmeal can be ground to make cornbread, polenta, and corn tortillas. Corn provides antioxidants, minerals, and vitamins in moderation. Most synthetic Vitamin C comes from GMO maize.
Corn oil, corn starch, dextrose (a sugar), and high-fructose corn syrup are often overlooked. They're stealth corn because they sneak into practically everything. Corn oil is used for frying, baking, and in potato chips, mayonnaise, margarine, and salad dressing. Baby food, bread, cakes, antibiotics, canned vegetables, beverages, and even dairy and animal products include corn starch. Dextrose appears in almost all prepared foods, excluding those with high-fructose corn syrup. HFCS isn't as easily digested as sucrose (from cane sugar). It can also cause other ailments, which we'll discuss later.
Most foods contain corn. It's fed to almost all food animals. 96% of U.S. animal feed is corn. 39% of U.S. corn is fed to livestock. But animals prefer other foods. Omnivore chickens prefer insects, worms, grains, and grasses. Captive cows are fed a total mixed ration, which contains corn. These animals' products, like eggs and milk, are also corn-fed.
There are numerous non-edible by-products of corn that are employed in the production of items like:
fuel-grade ethanol
plastics
batteries
cosmetics
meds/vitamins binder
carpets, fabrics
glutathione
crayons
Paint/glue
How does corn influence you? Consider quick food for dinner. You order a cheeseburger, fries, and big Coke at the counter (or drive-through in the suburbs). You tell yourself, "No corn." All that contains corn. Deconstruct:
Cows fed corn produce meat and cheese. Meat and cheese were bonded with corn syrup and starch (same). The bun (corn flour and dextrose) and fries were fried in maize oil. High fructose corn syrup sweetens the drink and helps make the cup and straw.
Just about everything contains corn. Then what? A cornspiracy, perhaps? Is eating too much maize an issue, or should we strive to stay away from it whenever possible?
As I've said, eating some maize can be healthy. 92% of U.S. corn is genetically modified, according to the Center for Food Safety. The adjustments are expected to boost corn yields. Some sweet corn is genetically modified to produce its own insecticide, a protein deadly to insects made by Bacillus thuringiensis. It's safe to eat in sweet corn. Concerns exist about feeding agricultural animals so much maize, modified or not.
High fructose corn syrup should be consumed in moderation. Fructose, a sugar, isn't easily metabolized. Fructose causes diabetes, fatty liver, obesity, and heart disease. It causes inflammation, which might aggravate gout. Candy, packaged sweets, soda, fast food, juice drinks, ice cream, ice cream topping syrups, sauces & condiments, jams, bread, crackers, and pancake syrup contain the most high fructose corn syrup. Everyday foods with little nutrients. Check labels and choose cane sugar or sucrose-sweetened goods. Or, eat corn like the Corn Kid.

Alexandra Walker-Jones
3 years ago
These are the 15 foods you should eat daily and why.
Research on preventing disease, extending life, and caring for your body from the inside out
Grapefruit and pomegranates aren't on the list, so ignore that. Mostly, I enjoyed the visual, but those fruits are healthful, too.
15 (or 17 if you consider the photo) different foods a day sounds like a lot. If you're not used to it — it is.
These lists don't aim for perfection. Instead, use this article and the science below to eat more of these foods. If you can eat 5 foods one day and 5 the next, you're doing well. This list should be customized to your requirements and preferences.
“Every time you eat or drink, you are either feeding disease or fighting it” -Heather Morgan.
The 15 Foods That You Should Consume Daily and Why:
1. Dark/Red Berries
(blueberries, blackberries, acai, goji, cherries, strawberries, raspberries)
The 2010 Global Burden of Disease Study is the greatest definitive analysis of death and disease risk factors in history. They found the primary cause of both death, disability, and disease inside the United States was diet.
Not eating enough fruit, and specifically berries, was one of the best predictors of disease (1).
What's special about berries? It's their color! Berries have the most antioxidants of any fruit, second only to spices. The American Cancer Society found that those who ate the most berries were less likely to die of cardiovascular disease.
2. Beans
Soybeans, black beans, kidney beans, lentils, split peas, chickpeas.
Beans are one of the most important predictors of survival in older people, according to global research (2).
For every 20 grams (2 tablespoons) of beans consumed daily, the risk of death is reduced by 8%.
Soybeans and soy foods are high in phytoestrogen, which reduces breast and prostate cancer risks. Phytoestrogen blocks the receptors' access to true estrogen, mitigating the effects of weight gain, dairy (high in estrogen), and hormonal fluctuations (3).
3. Nuts
(almonds, walnuts, pecans, pistachios, Brazil nuts, cashews, hazelnuts, macadamia nuts)
Eating a handful of nuts every day reduces the risk of chronic diseases like heart disease and diabetes. Nuts also reduce oxidation, blood sugar, and LDL (bad) cholesterol, improving arterial function (4).
Despite their high-fat content, studies have linked daily nut consumption to a slimmer waistline and a lower risk of obesity (5).
4. Flaxseed
(milled flaxseed)
2013 research found that ground flaxseed had one of the strongest anti-hypertensive effects of any food. A few tablespoons (added to a smoothie or baked goods) lowered blood pressure and stroke risk 23 times more than daily aerobic exercise (6).
Flax shouldn't replace exercise, but its nutritional punch is worth adding to your diet.
5. Other seeds
(chia seeds, hemp seeds, pumpkin seeds, sesame seeds, fennel seeds)
Seeds are high in fiber and omega-3 fats and can be added to most dishes without being noticed.
When eaten with or after a meal, chia seeds moderate blood sugar and reduce inflammatory chemicals in the blood (7). Overall, a great daily addition.
6. Dates
Dates are one of the world's highest sugar foods, with 80% sugar by weight. Pure cake frosting is 60%, maple syrup is 66%, and cotton-candy jelly beans are 70%.
Despite their high sugar content, dates have a low glycemic index, meaning they don't affect blood sugar levels dramatically. They also improve triglyceride and antioxidant stress levels (8).
Dates are a great source of energy and contain high levels of dietary fiber and polyphenols, making 3-10 dates a great way to fight disease, support gut health with prebiotics, and satisfy a sweet tooth (9).
7. Cruciferous Veggies
(broccoli, Brussel sprouts, horseradish, kale, cauliflower, cabbage, boy choy, arugula, radishes, turnip greens)
Cruciferous vegetables contain an active ingredient that makes them disease-fighting powerhouses. Sulforaphane protects our brain, eyesight, against free radicals and environmental hazards, and treats and prevents cancer (10).
Unless you eat raw cruciferous vegetables daily, you won't get enough sulforaphane (and thus, its protective nutritional benefits). Cooking destroys the enzyme needed to create this super-compound.
If you chop broccoli, cauliflower, or turnip greens and let them sit for 45 minutes before cooking them, the enzyme will have had enough time to work its sulforaphane magic, allowing the vegetables to retain the same nutritional value as if eaten raw. Crazy, right? For more on this, see What Chopping Your Vegetables Has to Do with Fighting Cancer.
8. Whole grains
(barley, brown rice, quinoa, oats, millet, popcorn, whole-wheat pasta, wild rice)
Whole-grains are one of the healthiest ways to consume your daily carbs and help maintain healthy gut flora.
This happens when fibre is broken down in the colon and starts a chain reaction, releasing beneficial substances into the bloodstream and reducing the risk of Type 2 Diabetes and inflammation (11).
9. Spices
(turmeric, cumin, cinnamon, ginger, saffron, cloves, cardamom, chili powder, nutmeg, coriander)
7% of a person's cells will have DNA damage. This damage is caused by tiny breaks in our DNA caused by factors like free-radical exposure.
Free radicals cause mutations that damage lipids, proteins, and DNA, increasing the risk of disease and cancer. Free radicals are unavoidable because they result from cellular metabolism, but they can be avoided by consuming anti-oxidant and detoxifying foods.
Including spices and herbs like rosemary or ginger in our diet may cut DNA damage by 25%. Yes, this damage can be improved through diet. Turmeric worked better at a lower dose (just a pinch, daily). For maximum free-radical fighting (and anti-inflammatory) effectiveness, use 1.5 tablespoons of similar spices (12).
10. Leafy greens
(spinach, collard greens, lettuce, other salad greens, swiss chard)
Studies show that people who eat more leafy greens perform better on cognitive tests and slow brain aging by a year or two (13).
As we age, blood flow to the brain drops due to a decrease in nitric oxide, which prevents blood vessels from dilatation. Daily consumption of nitrate-rich vegetables like spinach and swiss chard may prevent dementia and Alzheimer's.
11. Fermented foods
(sauerkraut, tempeh, kombucha, plant-based kefir)
Miso, kimchi, and sauerkraut contain probiotics that support gut microbiome.
Probiotics balance the good and bad bacteria in our bodies and offer other benefits. Fermenting fruits and vegetables increases their antioxidant and vitamin content, preventing disease in multiple ways (14).
12. Sea vegetables
(seaweed, nori, dulse flakes)
A population study found that eating one sheet of nori seaweed per day may cut breast cancer risk by more than half (15).
Seaweed and sea vegetables may help moderate estrogen levels in the metabolism, reducing cancer and disease risk.
Sea vegetables make up 30% of the world's edible plants and contain unique phytonutrients. A teaspoon of these super sea-foods on your dinner will help fight disease from the inside out.
13. Water
I'm less concerned about whether you consider water food than whether you drink enough. If this list were ranked by what single item led to the best health outcomes, water would be first.
Research shows that people who drink 5 or more glasses of water per day have a 50% lower risk of dying from heart disease than those who drink 2 or less (16).
Drinking enough water boosts energy, improves skin, mental health, and digestion, and reduces the risk of various health issues, including obesity.
14. Tea
All tea consumption is linked to a lower risk of stroke, heart disease, and early death, with green tea leading for antioxidant content and immediate health benefits.
Green tea leaves may also be able to interfere with each stage of cancer formation, from the growth of the first mutated cell to the spread and progression of cancer in the body. Green tea is a quick and easy way to support your long-term and short-term health (17).
15. Supplemental B12 vitamin
B12, or cobalamin, is a vitamin responsible for cell metabolism. Not getting enough B12 can have serious consequences.
Historically, eating vegetables from untreated soil helped humans maintain their vitamin B12 levels. Due to modern sanitization, our farming soil lacks B12.
B12 is often cited as a problem only for vegetarians and vegans (as animals we eat are given B12 supplements before slaughter), but recent studies have found that plant-based eaters have lower B12 deficiency rates than any other diet (18).
Article Sources:
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Jari Roomer
3 years ago
10 Alternatives to Smartphone Scrolling
"Don't let technology control you; manage your phone."
"Don't become a slave to technology," said Richard Branson. "Manage your phone, don't let it manage you."
Unfortunately, most people are addicted to smartphones.
Worrying smartphone statistics:
46% of smartphone users spend 5–6 hours daily on their device.
The average adult spends 3 hours 54 minutes per day on mobile devices.
We check our phones 150–344 times per day (every 4 minutes).
During the pandemic, children's daily smartphone use doubled.
Having a list of productive, healthy, and fulfilling replacement activities is an effective way to reduce smartphone use.
The more you practice these smartphone replacements, the less time you'll waste.
Skills Development
Most people say they 'don't have time' to learn new skills or read more. Lazy justification. The issue isn't time, but time management. Distractions and low-quality entertainment waste hours every day.
The majority of time is spent in low-quality ways, according to Richard Koch, author of The 80/20 Principle.
What if you swapped daily phone scrolling for skill-building?
There are dozens of skills to learn, from high-value skills to make more money to new languages and party tricks.
Learning a new skill will last for years, if not a lifetime, compared to scrolling through your phone.
Watch Docs
Love documentaries. It's educational and relaxing. A good documentary helps you understand the world, broadens your mind, and inspires you to change.
Recent documentaries I liked include:
14 Peaks: Nothing Is Impossible
The Social Dilemma
Jim & Andy: The Great Beyond
Fantastic Fungi
Make money online
If you've ever complained about not earning enough money, put away your phone and get to work.
Instead of passively consuming mobile content, start creating it. Create something worthwhile. Freelance.
Internet makes starting a business or earning extra money easier than ever.
(Grand)parents didn't have this. Someone made them work 40+ hours. Few alternatives existed.
Today, all you need is internet and a monetizable skill. Use the internet instead of letting it distract you. Profit from it.
Bookworm
Jack Canfield, author of Chicken Soup For The Soul, said, "Everyone spends 2–3 hours a day watching TV." If you read that much, you'll be in the top 1% of your field."
Few people have more than two hours per day to read.
If you read 15 pages daily, you'd finish 27 books a year (as the average non-fiction book is about 200 pages).
Jack Canfield's quote remains relevant even though 15 pages can be read in 20–30 minutes per day. Most spend this time watching TV or on their phones.
What if you swapped 20 minutes of mindless scrolling for reading? You'd gain knowledge and skills.
Favorite books include:
The 7 Habits of Highly Effective People — Stephen R. Covey
The War of Art — Steven Pressfield
The Psychology of Money — Morgan Housel
A New Earth — Eckart Tolle
Get Organized
All that screen time could've been spent organizing. It could have been used to clean, cook, or plan your week.
If you're always 'behind,' spend 15 minutes less on your phone to get organized.
"Give me six hours to chop down a tree, and I'll spend the first four sharpening the ax," said Abraham Lincoln. Getting organized is like sharpening an ax, making each day more efficient.
Creativity
Why not be creative instead of consuming others'? Do something creative, like:
Painting
Musically
Photography\sWriting
Do-it-yourself
Construction/repair
Creative projects boost happiness, cognitive functioning, and reduce stress and anxiety. Creative pursuits induce a flow state, a powerful mental state.
This contrasts with smartphones' effects. Heavy smartphone use correlates with stress, depression, and anxiety.
Hike
People spend 90% of their time indoors, according to research. This generation is the 'Indoor Generation'
We lack an active lifestyle, fresh air, and vitamin D3 due to our indoor lifestyle (generated through direct sunlight exposure). Mental and physical health issues result.
Put away your phone and get outside. Go on nature walks. Explore your city on foot (or by bike, as we do in Amsterdam) if you live in a city. Move around! Outdoors!
You can't spend your whole life staring at screens.
Podcasting
Okay, a smartphone is needed to listen to podcasts. When you use your phone to get smarter, you're more productive than 95% of people.
Favorite podcasts:
The Pomp Podcast (about cryptocurrencies)
The Joe Rogan Experience
Kwik Brain (by Jim Kwik)
Podcasts can be enjoyed while walking, cleaning, or doing laundry. Win-win.
Journalize
I find journaling helpful for mental clarity. Writing helps organize thoughts.
Instead of reading internet opinions, comments, and discussions, look inward. Instead of Twitter or TikTok, look inward.
“It never ceases to amaze me: we all love ourselves more than other people, but care more about their opinion than our own.” — Marcus Aurelius
Give your mind free reign with pen and paper. It will highlight important thoughts, emotions, or ideas.
Never write for another person. You want unfiltered writing. So you get the best ideas.
Find your best hobbies
List your best hobbies. I guarantee 95% of people won't list smartphone scrolling.
It's often low-quality entertainment. The dopamine spike is short-lived, and it leaves us feeling emotionally 'empty'
High-quality leisure sparks happiness. They make us happy and alive. Everyone has different interests, so these activities vary.
My favorite quality hobbies are:
Nature walks (especially the mountains)
Video game party
Watching a film with my girlfriend
Gym weightlifting
Complexity learning (such as the blockchain and the universe)
This brings me joy. They make me feel more fulfilled and 'rich' than social media scrolling.
Make a list of your best hobbies to refer to when you're spending too much time on your phone.

Tim Soulo
3 years ago
Here is why 90.63% of Pages Get No Traffic From Google.
The web adds millions or billions of pages per day.
How much Google traffic does this content get?
In 2017, we studied 2 million randomly-published pages to answer this question. Only 5.7% of them ranked in Google's top 10 search results within a year of being published.
94.3 percent of roughly two million pages got no Google traffic.
Two million pages is a small sample compared to the entire web. We did another study.
We analyzed over a billion pages to see how many get organic search traffic and why.
How many pages get search traffic?
90% of pages in our index get no Google traffic, and 5.2% get ten visits or less.
90% of google pages get no organic traffic
How can you join the minority that gets Google organic search traffic?
There are hundreds of SEO problems that can hurt your Google rankings. If we only consider common scenarios, there are only four.
Reason #1: No backlinks
I hate to repeat what most SEO articles say, but it's true:
Backlinks boost Google rankings.
Google's "top 3 ranking factors" include them.
Why don't we divide our studied pages by the number of referring domains?
66.31 percent of pages have no backlinks, and 26.29 percent have three or fewer.
Did you notice the trend already?
Most pages lack search traffic and backlinks.
But are these the same pages?
Let's compare monthly organic search traffic to backlinks from unique websites (referring domains):
More backlinks equals more Google organic traffic.
Referring domains and keyword rankings are correlated.
It's important to note that correlation does not imply causation, and none of these graphs prove backlinks boost Google rankings. Most SEO professionals agree that it's nearly impossible to rank on the first page without backlinks.
You'll need high-quality backlinks to rank in Google and get search traffic.
Is organic traffic possible without links?
Here are the numbers:
Four million pages get organic search traffic without backlinks. Only one in 20 pages without backlinks has traffic, which is 5% of our sample.
Most get 300 or fewer organic visits per month.
What happens if we exclude high-Domain-Rating pages?
The numbers worsen. Less than 4% of our sample (1.4 million pages) receive organic traffic. Only 320,000 get over 300 monthly organic visits, or 0.1% of our sample.
This suggests high-authority pages without backlinks are more likely to get organic traffic than low-authority pages.
Internal links likely pass PageRank to new pages.
Two other reasons:
Our crawler's blocked. Most shady SEOs block backlinks from us. This prevents competitors from seeing (and reporting) PBNs.
They choose low-competition subjects. Low-volume queries are less competitive, requiring fewer backlinks to rank.
If the idea of getting search traffic without building backlinks excites you, learn about Keyword Difficulty and how to find keywords/topics with decent traffic potential and low competition.
Reason #2: The page has no long-term traffic potential.
Some pages with many backlinks get no Google traffic.
Why? I filtered Content Explorer for pages with no organic search traffic and divided them into four buckets by linking domains.
Almost 70k pages have backlinks from over 200 domains, but no search traffic.
By manually reviewing these (and other) pages, I noticed two general trends that explain why they get no traffic:
They overdid "shady link building" and got penalized by Google;
They're not targeting a Google-searched topic.
I won't elaborate on point one because I hope you don't engage in "shady link building"
#2 is self-explanatory:
If nobody searches for what you write, you won't get search traffic.
Consider one of our blog posts' metrics:
No organic traffic despite 337 backlinks from 132 sites.
The page is about "organic traffic research," which nobody searches for.
News articles often have this. They get many links from around the web but little Google traffic.
People can't search for things they don't know about, and most don't care about old events and don't search for them.
Note:
Some news articles rank in the "Top stories" block for relevant, high-volume search queries, generating short-term organic search traffic.
The Guardian's top "Donald Trump" story:
Ahrefs caught on quickly:
"Donald Trump" gets 5.6M monthly searches, so this page got a lot of "Top stories" traffic.
I bet traffic has dropped if you check now.
One of the quickest and most effective SEO wins is:
Find your website's pages with the most referring domains;
Do keyword research to re-optimize them for relevant topics with good search traffic potential.
Bryan Harris shared this "quick SEO win" during a course interview:
He suggested using Ahrefs' Site Explorer's "Best by links" report to find your site's most-linked pages and analyzing their search traffic. This finds pages with lots of links but little organic search traffic.
We see:
The guide has 67 backlinks but no organic traffic.
We could fix this by re-optimizing the page for "SERP"
A similar guide with 26 backlinks gets 3,400 monthly organic visits, so we should easily increase our traffic.
Don't do this with all low-traffic pages with backlinks. Choose your battles wisely; some pages shouldn't be ranked.
Reason #3: Search intent isn't met
Google returns the most relevant search results.
That's why blog posts with recommendations rank highest for "best yoga mat."
Google knows that most searchers aren't buying.
It's also why this yoga mats page doesn't rank, despite having seven times more backlinks than the top 10 pages:
The page ranks for thousands of other keywords and gets tens of thousands of monthly organic visits. Not being the "best yoga mat" isn't a big deal.
If you have pages with lots of backlinks but no organic traffic, re-optimizing them for search intent can be a quick SEO win.
It was originally a boring landing page describing our product's benefits and offering a 7-day trial.
We realized the problem after analyzing search intent.
People wanted a free tool, not a landing page.
In September 2018, we published a free tool at the same URL. Organic traffic and rankings skyrocketed.
Reason #4: Unindexed page
Google can’t rank pages that aren’t indexed.
If you think this is the case, search Google for site:[url]. You should see at least one result; otherwise, it’s not indexed.
A rogue noindex meta tag is usually to blame. This tells search engines not to index a URL.
Rogue canonicals, redirects, and robots.txt blocks prevent indexing.
Check the "Excluded" tab in Google Search Console's "Coverage" report to see excluded pages.
Google doesn't index broken pages, even with backlinks.
Surprisingly common.
In Ahrefs' Site Explorer, the Best by Links report for a popular content marketing blog shows many broken pages.
One dead page has 131 backlinks:
According to the URL, the page defined content marketing. —a keyword with a monthly search volume of 5,900 in the US.
Luckily, another page ranks for this keyword. Not a huge loss.
At least redirect the dead page's backlinks to a working page on the same topic. This may increase long-tail keyword traffic.
This post is a summary. See the original post here

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
