Applications of polynomials of a high degree
What is the highest polynomial degree that has an application in real life, and what is that application? My google search yielded 3rd degree at most.
polynomials applications
asked Nov 30 at 14:36
gil_mo
62
add a comment |
What is the highest polynomial degree that has an application in real life, and what is that application? My google search yielded 3rd degree at most.
polynomials applications
asked Nov 30 at 14:36
gil_mo
62
3
This is a fuzzy question, depending on what you call real life. (Many people live without ever using polynomials.) In engineering and scientific circles, higher degree polynomials are certainly in use, with a frequency that decreases with the degree. Mathematicians can play with polynomials of thousands of terms... There is no "highest".
– Yves Daoust
Nov 30 at 14:44
Eigenvalue problems in physics can be (informally) considered to be searching for roots of infinite-degree polynomials. Guess it will be hard to find a finite upper bound.
– lisyarus
Nov 30 at 14:55
1
Stefan-Boltzmann's law (quartic) must b e a candidate for most frequent actual evaluation of higher-than-cubic polynomial outside formal polynomials and function expansions / transforms.
– Dannie
Nov 30 at 14:59
1
@dannie The expression of the Van der Walls forces has a factor in $R^7$.
– Yves Daoust
Nov 30 at 15:17
1
@YvesDaoust Lennard-Jones potential has even $R^{12}$.
– lisyarus
Nov 30 at 15:32
add a comment |
What is the highest polynomial degree that has an application in real life, and what is that application? My google search yielded 3rd degree at most.
polynomials applications
asked Nov 30 at 14:36
gil_mo
62
What is the highest polynomial degree that has an application in real life, and what is that application? My google search yielded 3rd degree at most.
polynomials applications
polynomials applications
asked Nov 30 at 14:36
gil_mo
62
asked Nov 30 at 14:36
gil_mo
62
asked Nov 30 at 14:36
gil_mo
62
asked Nov 30 at 14:36
gil_mo
62
asked Nov 30 at 14:36
gil_mo
62
62
3
This is a fuzzy question, depending on what you call real life. (Many people live without ever using polynomials.) In engineering and scientific circles, higher degree polynomials are certainly in use, with a frequency that decreases with the degree. Mathematicians can play with polynomials of thousands of terms... There is no "highest".
– Yves Daoust
Nov 30 at 14:44
Eigenvalue problems in physics can be (informally) considered to be searching for roots of infinite-degree polynomials. Guess it will be hard to find a finite upper bound.
– lisyarus
Nov 30 at 14:55
1
Stefan-Boltzmann's law (quartic) must b e a candidate for most frequent actual evaluation of higher-than-cubic polynomial outside formal polynomials and function expansions / transforms.
– Dannie
Nov 30 at 14:59
1
@dannie The expression of the Van der Walls forces has a factor in $R^7$.
– Yves Daoust
Nov 30 at 15:17
1
@YvesDaoust Lennard-Jones potential has even $R^{12}$.
– lisyarus
Nov 30 at 15:32
add a comment |
3
This is a fuzzy question, depending on what you call real life. (Many people live without ever using polynomials.) In engineering and scientific circles, higher degree polynomials are certainly in use, with a frequency that decreases with the degree. Mathematicians can play with polynomials of thousands of terms... There is no "highest".
– Yves Daoust
Nov 30 at 14:44
Eigenvalue problems in physics can be (informally) considered to be searching for roots of infinite-degree polynomials. Guess it will be hard to find a finite upper bound.
– lisyarus
Nov 30 at 14:55
1
Stefan-Boltzmann's law (quartic) must b e a candidate for most frequent actual evaluation of higher-than-cubic polynomial outside formal polynomials and function expansions / transforms.
– Dannie
Nov 30 at 14:59
1
@dannie The expression of the Van der Walls forces has a factor in $R^7$.
– Yves Daoust
Nov 30 at 15:17
1
@YvesDaoust Lennard-Jones potential has even $R^{12}$.
– lisyarus
Nov 30 at 15:32
3
3
This is a fuzzy question, depending on what you call real life. (Many people live without ever using polynomials.) In engineering and scientific circles, higher degree polynomials are certainly in use, with a frequency that decreases with the degree. Mathematicians can play with polynomials of thousands of terms... There is no "highest".
– Yves Daoust
Nov 30 at 14:44
This is a fuzzy question, depending on what you call real life. (Many people live without ever using polynomials.) In engineering and scientific circles, higher degree polynomials are certainly in use, with a frequency that decreases with the degree. Mathematicians can play with polynomials of thousands of terms... There is no "highest".
– Yves Daoust
Nov 30 at 14:44
Eigenvalue problems in physics can be (informally) considered to be searching for roots of infinite-degree polynomials. Guess it will be hard to find a finite upper bound.
– lisyarus
Nov 30 at 14:55
Eigenvalue problems in physics can be (informally) considered to be searching for roots of infinite-degree polynomials. Guess it will be hard to find a finite upper bound.
– lisyarus
Nov 30 at 14:55
1
1
Stefan-Boltzmann's law (quartic) must b e a candidate for most frequent actual evaluation of higher-than-cubic polynomial outside formal polynomials and function expansions / transforms.
– Dannie
Nov 30 at 14:59
Stefan-Boltzmann's law (quartic) must b e a candidate for most frequent actual evaluation of higher-than-cubic polynomial outside formal polynomials and function expansions / transforms.
– Dannie
Nov 30 at 14:59
1
1
@dannie The expression of the Van der Walls forces has a factor in $R^7$.
– Yves Daoust
Nov 30 at 15:17
@dannie The expression of the Van der Walls forces has a factor in $R^7$.
– Yves Daoust
Nov 30 at 15:17
1
1
@YvesDaoust Lennard-Jones potential has even $R^{12}$.
– lisyarus
Nov 30 at 15:32
@YvesDaoust Lennard-Jones potential has even $R^{12}$.
– lisyarus
Nov 30 at 15:32
add a comment |
2 Answers
2
active
oldest
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Practitioners in signal and image processing heavily use the so-called Discrete Fourier Transform, which is a polynomial evaluated on complex variables. Applications in medical imaging abound.
The degree of these polynomials typically reaches the image size, which can be like 4096 or much more.
Other consumers of polynomials are the error correction methods (such as those used in digital communication or CD readers). They are implemented in some types of barcodes, with degrees that reach dozens. If I am right, the QR codes use up to degree 69.
answered Nov 30 at 14:51
Yves Daoust
124k671221
A good one. IIRC the error correcting code used in CDs uses polynomials up to degree 28.
– Jyrki Lahtonen
Nov 30 at 15:12
add a comment |
The characteristic polynomial of any matrix (though rarely explicitly computed) gives quite a bit of information about that matrix that is incredibly useful in a huge variety of data analysis settings (machine learning, dimensionality reduction, etc.) The degree of this polynomial is the same as the size of the matrix (depending on setting, this can easily be in the multiple thousands, or even larger). Admittedly this is more of a computer science answer than math, but is still relevant.
answered Nov 30 at 14:54
DreamConspiracy
8901216
add a comment |
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2 Answers
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active
oldest
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2 Answers
2
active
oldest
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active
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Practitioners in signal and image processing heavily use the so-called Discrete Fourier Transform, which is a polynomial evaluated on complex variables. Applications in medical imaging abound.
The degree of these polynomials typically reaches the image size, which can be like 4096 or much more.
Other consumers of polynomials are the error correction methods (such as those used in digital communication or CD readers). They are implemented in some types of barcodes, with degrees that reach dozens. If I am right, the QR codes use up to degree 69.
answered Nov 30 at 14:51
Yves Daoust
124k671221
A good one. IIRC the error correcting code used in CDs uses polynomials up to degree 28.
– Jyrki Lahtonen
Nov 30 at 15:12
add a comment |
Practitioners in signal and image processing heavily use the so-called Discrete Fourier Transform, which is a polynomial evaluated on complex variables. Applications in medical imaging abound.
The degree of these polynomials typically reaches the image size, which can be like 4096 or much more.
Other consumers of polynomials are the error correction methods (such as those used in digital communication or CD readers). They are implemented in some types of barcodes, with degrees that reach dozens. If I am right, the QR codes use up to degree 69.
answered Nov 30 at 14:51
Yves Daoust
124k671221
A good one. IIRC the error correcting code used in CDs uses polynomials up to degree 28.
– Jyrki Lahtonen
Nov 30 at 15:12
add a comment |
Practitioners in signal and image processing heavily use the so-called Discrete Fourier Transform, which is a polynomial evaluated on complex variables. Applications in medical imaging abound.
The degree of these polynomials typically reaches the image size, which can be like 4096 or much more.
Other consumers of polynomials are the error correction methods (such as those used in digital communication or CD readers). They are implemented in some types of barcodes, with degrees that reach dozens. If I am right, the QR codes use up to degree 69.
answered Nov 30 at 14:51
Yves Daoust
124k671221
Practitioners in signal and image processing heavily use the so-called Discrete Fourier Transform, which is a polynomial evaluated on complex variables. Applications in medical imaging abound.
The degree of these polynomials typically reaches the image size, which can be like 4096 or much more.
Other consumers of polynomials are the error correction methods (such as those used in digital communication or CD readers). They are implemented in some types of barcodes, with degrees that reach dozens. If I am right, the QR codes use up to degree 69.
answered Nov 30 at 14:51
Yves Daoust
124k671221
edited Nov 30 at 15:08
answered Nov 30 at 14:51
Yves Daoust
124k671221
answered Nov 30 at 14:51
Yves Daoust
124k671221
answered Nov 30 at 14:51
Yves Daoust
124k671221
124k671221
A good one. IIRC the error correcting code used in CDs uses polynomials up to degree 28.
– Jyrki Lahtonen
Nov 30 at 15:12
add a comment |
A good one. IIRC the error correcting code used in CDs uses polynomials up to degree 28.
– Jyrki Lahtonen
Nov 30 at 15:12
A good one. IIRC the error correcting code used in CDs uses polynomials up to degree 28.
– Jyrki Lahtonen
Nov 30 at 15:12
A good one. IIRC the error correcting code used in CDs uses polynomials up to degree 28.
– Jyrki Lahtonen
Nov 30 at 15:12
add a comment |
The characteristic polynomial of any matrix (though rarely explicitly computed) gives quite a bit of information about that matrix that is incredibly useful in a huge variety of data analysis settings (machine learning, dimensionality reduction, etc.) The degree of this polynomial is the same as the size of the matrix (depending on setting, this can easily be in the multiple thousands, or even larger). Admittedly this is more of a computer science answer than math, but is still relevant.
answered Nov 30 at 14:54
DreamConspiracy
8901216
add a comment |
The characteristic polynomial of any matrix (though rarely explicitly computed) gives quite a bit of information about that matrix that is incredibly useful in a huge variety of data analysis settings (machine learning, dimensionality reduction, etc.) The degree of this polynomial is the same as the size of the matrix (depending on setting, this can easily be in the multiple thousands, or even larger). Admittedly this is more of a computer science answer than math, but is still relevant.
answered Nov 30 at 14:54
DreamConspiracy
8901216
add a comment |
The characteristic polynomial of any matrix (though rarely explicitly computed) gives quite a bit of information about that matrix that is incredibly useful in a huge variety of data analysis settings (machine learning, dimensionality reduction, etc.) The degree of this polynomial is the same as the size of the matrix (depending on setting, this can easily be in the multiple thousands, or even larger). Admittedly this is more of a computer science answer than math, but is still relevant.
answered Nov 30 at 14:54
DreamConspiracy
8901216
The characteristic polynomial of any matrix (though rarely explicitly computed) gives quite a bit of information about that matrix that is incredibly useful in a huge variety of data analysis settings (machine learning, dimensionality reduction, etc.) The degree of this polynomial is the same as the size of the matrix (depending on setting, this can easily be in the multiple thousands, or even larger). Admittedly this is more of a computer science answer than math, but is still relevant.
answered Nov 30 at 14:54
DreamConspiracy
8901216
answered Nov 30 at 14:54
DreamConspiracy
8901216
answered Nov 30 at 14:54
DreamConspiracy
8901216
answered Nov 30 at 14:54
DreamConspiracy
8901216
8901216
add a comment |
add a comment |
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3
This is a fuzzy question, depending on what you call real life. (Many people live without ever using polynomials.) In engineering and scientific circles, higher degree polynomials are certainly in use, with a frequency that decreases with the degree. Mathematicians can play with polynomials of thousands of terms... There is no "highest".
– Yves Daoust
Nov 30 at 14:44
Eigenvalue problems in physics can be (informally) considered to be searching for roots of infinite-degree polynomials. Guess it will be hard to find a finite upper bound.
– lisyarus
Nov 30 at 14:55
1
Stefan-Boltzmann's law (quartic) must b e a candidate for most frequent actual evaluation of higher-than-cubic polynomial outside formal polynomials and function expansions / transforms.
– Dannie
Nov 30 at 14:59
1
@dannie The expression of the Van der Walls forces has a factor in $R^7$.
– Yves Daoust
Nov 30 at 15:17
1
@YvesDaoust Lennard-Jones potential has even $R^{12}$.
– lisyarus
Nov 30 at 15:32